Your original if statement would have worked fine. I think it's your SQL connection and/or statement. You should try to qualify those single quotes in that SQL statement.
I.E. instead of:
where Member_ID='$appUsername'
try:
where Member_ID=\'$appUsername\'
And do the same for the Cust_ID.
Just IMHO, Cust_ID is a terrible column name for the password that's associated with Member_ID :-) Make it Member_Pass or something that is relevant.
If the above suggestion doesn't work, let me know.