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There are a couple of issues in your code:

• The #!-line should read #!/bin/bash (or whatever the full absolute path is to the bash interpreter or your system). Note that you are using shell code that requires the use of bash, so pointing the #!-line to the sh interpreter would not be correct as that shell may not understand things like the non-standard [[ ... ]], or the use of arrays.

• Passwords should probably be read using IFS= read -s -r in bash to avoid echoing the input to the terminal, and to allow the use of \ without issues, and to allow flanking spaces etc.

• Multiple tests are better written [ ... ] || [ ... ]. This is what causes your actual error message. The [ utility does not know about ||.

Suggestion:

#!/bin/bash

IFS= read -p 'Enter password: ' -sr password
check=(0 0)

[[ $password =~ [A-Z] ]] && check[0]=1
[[ $password =~ [a-z] ]] && check[1]=1

if [ "${check[0]}" = 1 ] || [ "${check[1]}" = 1 ]; then
    echo yes
else
    echo no
fi

Or, getting rid of the need for the array:

#!/bin/bash

IFS= read -p 'Enter password: ' -sr password

if [[ $password =~ [A-Z] ]] ||
   [[ $password =~ [a-z] ]]
then
    echo yes
else
    echo no
fi

Note that testing the $password string for matches of [A-Z] or for matches of [a-z] can be done in one test with [A-Za-z].

In the sh shell, you could write your script like this:

#!/bin/sh

printf 'Enter password: ' >&2

state=$(stty -g)    # save stty state
stty -echo          # turn off local echo
IFS= read -r password
stty "$state"       # reset stty state to original

case $password in
        *[A-Za-z]*)
                echo yes
                ;;
        *)
                echo no
esac

Note that I've combined the tests for upper and lower case characters into a single test, and that the patterns used by case are shell globbing patterns, not regular expressions.