The example below is a copy of the Custom Pagination Actions example from the documentation, but with the style change indicated in your question.
The changes I have done to the demo are adding your styles for the Paper background and a style for the MenuItem hover effect (otherwise the hover isn't visible):
const useStyles2 = makeStyles({
selectDropdown: { color: "#fff", backgroundColor: "#1b1f38" },
menuItem: {
"&:hover": {
backgroundColor: "#3b3f58"
}
}
});
To apply your selectDropdown class to the menu Paper, you use the MenuProps within the SelectProps. The menuItem style can be applied more directly (both shown below).
<TablePagination
SelectProps={{
MenuProps: { classes: { paper: classes.selectDropdown } }
}}
classes={{ menuItem: classes.menuItem }}
...
/>
The other change, compared to the original documentation demo, is that I removed native: true from the SelectProps (none of this custom styling approach is applicable for native select, and the example in your question is not using a native select).
Here's the full code of the sandbox:
import React from "react";
import PropTypes from "prop-types";
import { makeStyles, useTheme } from "@material-ui/core/styles";
import Table from "@material-ui/core/Table";
import TableBody from "@material-ui/core/TableBody";
import TableCell from "@material-ui/core/TableCell";
import TableContainer from "@material-ui/core/TableContainer";
import TableFooter from "@material-ui/core/TableFooter";
import TablePagination from "@material-ui/core/TablePagination";
import TableRow from "@material-ui/core/TableRow";
import Paper from "@material-ui/core/Paper";
import IconButton from "@material-ui/core/IconButton";
import FirstPageIcon from "@material-ui/icons/FirstPage";
import KeyboardArrowLeft from "@material-ui/icons/KeyboardArrowLeft";
import KeyboardArrowRight from "@material-ui/icons/KeyboardArrowRight";
import LastPageIcon from "@material-ui/icons/LastPage";
const useStyles1 = makeStyles(theme => ({
root: {
flexShrink: 0,
marginLeft: theme.spacing(2.5)
}
}));
function TablePaginationActions(props) {
const classes = useStyles1();
const theme = useTheme();
const { count, page, rowsPerPage, onChangePage } = props;
const handleFirstPageButtonClick = event => {
onChangePage(event, 0);
};
const handleBackButtonClick = event => {
onChangePage(event, page - 1);
};
const handleNextButtonClick = event => {
onChangePage(event, page + 1);
};
const handleLastPageButtonClick = event => {
onChangePage(event, Math.max(0, Math.ceil(count / rowsPerPage) - 1));
};
return (
<div className={classes.root}>
<IconButton
onClick={handleFirstPageButtonClick}
disabled={page === 0}
aria-label="first page"
>
{theme.direction === "rtl" ? <LastPageIcon /> : <FirstPageIcon />}
</IconButton>
<IconButton
onClick={handleBackButtonClick}
disabled={page === 0}
aria-label="previous page"
>
{theme.direction === "rtl" ? (
<KeyboardArrowRight />
) : (
<KeyboardArrowLeft />
)}
</IconButton>
<IconButton
onClick={handleNextButtonClick}
disabled={page >= Math.ceil(count / rowsPerPage) - 1}
aria-label="next page"
>
{theme.direction === "rtl" ? (
<KeyboardArrowLeft />
) : (
<KeyboardArrowRight />
)}
</IconButton>
<IconButton
onClick={handleLastPageButtonClick}
disabled={page >= Math.ceil(count / rowsPerPage) - 1}
aria-label="last page"
>
{theme.direction === "rtl" ? <FirstPageIcon /> : <LastPageIcon />}
</IconButton>
</div>
);
}
TablePaginationActions.propTypes = {
count: PropTypes.number.isRequired,
onChangePage: PropTypes.func.isRequired,
page: PropTypes.number.isRequired,
rowsPerPage: PropTypes.number.isRequired
};
function createData(name, calories, fat) {
return { name, calories, fat };
}
const rows = [
createData("Cupcake", 305, 3.7),
createData("Donut", 452, 25.0),
createData("Eclair", 262, 16.0),
createData("Frozen yoghurt", 159, 6.0),
createData("Gingerbread", 356, 16.0),
createData("Honeycomb", 408, 3.2),
createData("Ice cream sandwich", 237, 9.0),
createData("Jelly Bean", 375, 0.0),
createData("KitKat", 518, 26.0),
createData("Lollipop", 392, 0.2),
createData("Marshmallow", 318, 0),
createData("Nougat", 360, 19.0),
createData("Oreo", 437, 18.0)
].sort((a, b) => (a.calories < b.calories ? -1 : 1));
const useStyles2 = makeStyles({
table: {
minWidth: 500
},
selectDropdown: { color: "#fff", backgroundColor: "#1b1f38" },
menuItem: {
"&:hover": {
backgroundColor: "#3b3f58"
}
}
});
export default function CustomPaginationActionsTable() {
const classes = useStyles2();
const [page, setPage] = React.useState(0);
const [rowsPerPage, setRowsPerPage] = React.useState(5);
const emptyRows =
rowsPerPage - Math.min(rowsPerPage, rows.length - page * rowsPerPage);
const handleChangePage = (event, newPage) => {
setPage(newPage);
};
const handleChangeRowsPerPage = event => {
setRowsPerPage(parseInt(event.target.value, 10));
setPage(0);
};
return (
<TableContainer component={Paper}>
<Table className={classes.table} aria-label="custom pagination table">
<TableBody>
{(rowsPerPage > 0
? rows.slice(page * rowsPerPage, page * rowsPerPage + rowsPerPage)
: rows
).map(row => (
<TableRow key={row.name}>
<TableCell component="th" scope="row">
{row.name}
</TableCell>
<TableCell style={{ width: 160 }} align="right">
{row.calories}
</TableCell>
<TableCell style={{ width: 160 }} align="right">
{row.fat}
</TableCell>
</TableRow>
))}
{emptyRows > 0 && (
<TableRow style={{ height: 53 * emptyRows }}>
<TableCell colSpan={6} />
</TableRow>
)}
</TableBody>
<TableFooter>
<TableRow>
<TablePagination
rowsPerPageOptions={[5, 10, 25, { label: "All", value: -1 }]}
colSpan={3}
count={rows.length}
rowsPerPage={rowsPerPage}
page={page}
SelectProps={{
inputProps: { "aria-label": "rows per page" },
MenuProps: { classes: { paper: classes.selectDropdown } }
}}
classes={{ menuItem: classes.menuItem }}
onChangePage={handleChangePage}
onChangeRowsPerPage={handleChangeRowsPerPage}
ActionsComponent={TablePaginationActions}
/>
</TableRow>
</TableFooter>
</Table>
</TableContainer>
);
}

Related (also styles the Menu Paper) answer: (material-ui) Apply max-height to <Select> children
Related documentation: