In bash version 3 you can use the '=~' operator:

if [[ "$date" =~ ^[0-9]{8}$ ]]; then
    echo "Valid date"
else
    echo "Invalid date"
fi

Reference: http://tldp.org/LDP/abs/html/bashver3.html#REGEXMATCHREF

NOTE: The quoting in the matching operator within the double brackets, [[ ]], is no longer necessary as of Bash version 3.2

A good way to test if a string is a correct date is to use the command date:

if date -d "${DATE}" >/dev/null 2>&1
then
  # do what you need to do with your date
else
  echo "${DATE} incorrect date" >&2
  exit 1
fi

from comment: one can use formatting

if [ "2017-01-14" == $(date -d "2017-01-14" '+%Y-%m-%d') ] 

In addition to other answers of the =~ Bash operator - Extended Regular Expressions (ERE).

This is the syntax used by awk and egrep (or grep -E),
as well as by Bash's [[ ... =~ ... ]] operator.

For example, a function which supports multiple test provided in multiple arguments:

#!/bin/bash

#-----------#
# Functions #
#-----------#

function RT
{
    declare __line;

    for __line in "${@:2}";
    do
        if ! [[ "$__line" =~ $1 ]];
        then
            return 1;
        fi
    done

    return 0;
}

#-----------#
# Main      #
#-----------#

regex_v='^[0-9]*$';
value_1_v='12345';
value_2_v='67890';

if RT "$regex_v" "$value_1_v" "$value_2_v";
then
    printf 'Valid';
else
    printf 'Invalid';
fi

Description

Function RT or Regex Test

# Declare a local variable for a loop.

declare __line;

# Loop for every argument's value except the first - regex rule

for __line in "${@:2}";

# Test the value and return a **non-zero** return code if failed.
# Alternative: if [[ ! "$__line" =~ $1 ]];

if ! [[ "$__line" =~ $1 ]];

# Return a **zero** return code - success.

return 0;

Main code

# Define arguments for the function to test

regex_v='^[0-9]*$'; # Regex rule
value_1_v='12345'; # First value
value_2_v='67890'; # Second value

# A statement which runs the function with specified arguments
# and executes `printf 'Valid';` if succeeded, else - `printf 'Invalid';`

if RT "$regex_v" "$value_v";

It should be possible to point at failed argument, for example, by appending a counter to the loop and printing its value to stderr.

Related

The quotes around the right-hand side of the =~ operator cause it to become a string, rather than a RegularExpression.

Source

Where the usage of a regex can be helpful to determine if the character sequence of a date is correct, it cannot be used easily to determine if the date is valid. The following examples will pass the regular expression, but are all invalid dates: 20180231, 20190229, 20190431

So if you want to validate if your date string (let's call it datestr) is in the correct format, it is best to parse it with date and ask date to convert the string to the correct format. If both strings are identical, you have a valid format and valid date.

if [[ "$datestr" == $(date -d "$datestr" "+%Y%m%d" 2>/dev/null) ]]; then
     echo "Valid date"
else
     echo "Invalid date"
fi