This is an old question, but it is still valid and I think there is more to usefully, but briefly, say on the subject than is covered by existing answers.

Unicode

Unicode defines properties for characters.

One of these properties is "General Category" which has Major classes and subclasses. The Major classes are Letter, Mark, Punctuation, Symbol, Separator, and Other.

By knowing the properties of your characters, you can decide whether you consider them printable in your particular context.

You must always remember that terms like "character" and "printable" are often difficult and have interesting edge-cases.

Programming Language support

Some programming languages assist with this problem.

For example, the Go language has a "unicode" package which provides many useful Unicode-related functions including these two:

func IsGraphic(r rune) bool

IsGraphic reports whether the rune is defined as a Graphic by Unicode. Such  
characters include letters, marks, numbers, punctuation, symbols, and spaces, 
from categories L, M, N, P, S, Zs. 

func IsPrint(r rune) bool

IsPrint reports whether the rune is defined as printable by Go. Such  
characters include letters, marks, numbers, punctuation, symbols, and  
the ASCII space character, from categories L, M, N, P, S and the ASCII  
space character. This categorization is the same as IsGraphic except  
that the only spacing character is ASCII space, U+0020.

Notice that it says "defined as printable by Go" not by "defined as printable by Unicode". It is almost as if there are some depths the wizards at Unicode dare not plumb.

Printable

The more you learn about Unicode, the more you realise how unexpectedly diverse and unfathomably weird human writing systems are.

In particular whether a particular "character" is printable is not always obvious.

Is a zero-width space printable? When is a hyphenation point printable? Are there characters whose printability depends on their position in a word or on what characters are adjacent to them? Is a combining-character always printable?

Footnotes

ASCII printable character range is \u0020 - \u007f

No it isn't. \u007f is DEL which is not normally considered a printable character. It is, for example, associated with the keyboard key labelled "DEL" whose earliest purpose was to command the deletion of a character from some medium (display, file etc).

In fact many 8-bit character sets have many non-consecutive ranges which are non-printable. See for example C0 and C1 controls.

Taking the opposite approach to @HoldOffHunger, it might be easier to list the ranges of non-printable characters, and use not to test if a character is printable.

In the style of Regex (so if you wanted printable characters, place a ^):

[\u0000-\u0008\u000B-\u001F\u007F-\u009F\u2000-\u200F\u2028-\u202F\u205F-\u206F\u3000\uFEFF]

Which accounts for things like separator spaces and joiners

Note that unlike their answer which is a whitelist that ignores all non-latin languages, this blacklist wont permit non-printable characters just because they're in blocks with printable characters (their answer wholly includes Non-Latin, Language Supplement blocks as 'printable', even though it contains things like 'zero-width non-joiner'..).

Be aware though, that if using this or any other solution, for sanitation for example, you may want to do something more nuanced than a blanket replace.
Arguably in that case, non-breaking spaces should change to space, not be removed, and invisible separator should be replaced with comma conditionally.

Then there's invalid character ranges, either [yet] unused or reserved for encoding purposes, and language-specific variation selectors..

NB when using regular expressions, that you enable unicode awareness if it isn't that way by default (for javascript it's via /.../u).

You can tell if you have it correct by attempting to create the regular expression with some multi-byte character ranges.
For example, the above, plus the invalid character range \u{E0100}-\u{E01EF} in javascript:

/[\u0000-\u0008\u000B-\u001F\u007F-\u009F\u2000-\u200F\u2028-\u202F\u205F-\u206F\u3000\uFEFF\u{E0100}-\u{E01EF}]/u

Without u \u{E0100}-\u{E01EF} equates to \uDB40(\uDD00-\uDB40)\uDDEF, not (\uDB40\uDD00)-(\uDB40\uDDEF), and if replacing you should always enable u even when not including multbyte unicode in the regex itself as you might break surrogate pairs that exist in the text.

What characters are valid?

At present, Unicode is defined as starting from U+0000 and ending at U+10FFFF. The first block, Basic Latin, spans U+0000 to U+007F and the last block, Supplementary Private Use Area-B, spans U+100000 to 10FFFF. If you want to see all of these blocks, see here: Wikipedia.org: Unicode Block; List of Blocks.

Let's break down what's valid/invalid in the Latin Block1.

The Latin Block: TLDR

If you're interested in filtering out either invisible characters, you'll want to filter out:

• U+0000 to U+0008: Control
• U+000E to U+001F: Device (i.e., Control)
• U+007F: Delete (Control)
• U+008D to U+009F: Device (i.e., Control)

The Latin Block: Full Ranges

Here's the Latin block, broken up into smaller sections...

• U+0000 to U+0008: Control
• U+0009 to U+000C: Space
• U+000E to U+001F: Device (i.e., Control)
• U+0020: Space
• U+0021 to U+002F: Symbols
• U+0030 to U+0039: Numbers
• U+003A to U+0040: Symbols
• U+0041 to U+005A: Uppercase Letters
• U+005B to U+0060: Symbols
• U+0061 to U+007A: Lowercase Letters
• U+007B to U+007E: Symbols
• U+007F: Delete (Control)
• U+0080 to U+008C: Latin1-Supplement symbols.
• U+008D to U+009F: Device (i.e., Control)
• U+00A0: Non-breaking space. (i.e.,  )
• U+00A1 to U+00BF: Symbols.
• U+00C0 to U+00FF: Accented characters.

The Other Blocks

Unicode is famous for supporting non-Latin character sets, so what are these other blocks? This is just a broad overview, see the wikipedia.org page for the full, complete list.

Latin1 & Latin1-Related Blocks

• U+0000 to U+007F : Basic Latin
• U+0080 to U+00FF : Latin-1 Supplement
• U+0100 to U+017F : Latin Extended-A
• U+0180 to U+024F : Latin Extended-B

Combinable blocks

U+0250 to U+036F: 3 Blocks.

Non-Latin, Language blocks

U+0370 to U+1C7F: 55 Blocks.

Non-Latin, Language Supplement blocks

U+1C80 to U+209F: 11 Blocks.

Symbol blocks

U+20A0 to U+2BFF: 22 Blocks.

Ancient Language blocks

U+2C00 to U+2C5F: 1 Block (Glagolitic).

Language Extensions blocks

U+2C60 to U+FFEF: 66 Blocks.

Special blocks

U+FFF0 to U+FFFF: 1 Block (Specials).

One approach is to render each character to a texture and manually check if it is visible. This solution excludes spaces.

I've written such a program and used it to determine there are roughly 467241 printable characters within the first 471859 code points. I've selected this number because it covers all of the first 4 Planes of Unicode, which seem to contain all printable characters. See https://en.wikipedia.org/wiki/Plane_(Unicode)

I would much like to refine my program to produce the list of ranges, but for now here's what I am working with for anyone who needs immediate answers:

https://editor.p5js.org/SamyBencherif/sketches/_OE8Y3kS9

I am posting this tool because I think this question attracts a lot of people who are looking for slightly different applications of knowing printable ranges. Hopefully this is useful, even though it does not fully answer the question.

The printable Unicode character range, excluding the hex, is 32 to 126 in the int datatype.