How can I tell if a library was compiled with -g?

If you're running on Linux, use objdump --debugging. There should be an entry for each object file in the library. For object files without debugging symbols, you'll see something like:

objdump --debugging libvoidincr.a
In archive libvoidincr.a:

voidincr.o:     file format elf64-x86-64

If there are debugging symbols, the output will be much more verbose.

nm -a <lib> will print all symbols from library, including debug ones.

So you can compare the outputs of nm <lib> and nm -a <lib> - if they differ, your lib contains some debug symbols.

You can use objdump for this.

EDIT: From the man-page:

-W
--dwarf
Displays  the  contents of the DWARF debug sections in the file, if
any are present.

https://access.redhat.com/documentation/en-us/red_hat_enterprise_linux/6/html/developer_guide/debugging

The command readelf -wi file is a good verification of debuginfo, compiled within your program.