The curl command is enclosed in single-quotes, which disables the special meanings of all other characters, including " (and of course $). You can either "un-enclose" the variables (and re-enclose them in double-quotes if necessary), or else use double-quotes on the outside, and escape the internal double-quotes.
basic_password=$(ssh rsync@some_test_domain 'curl --silent "http://some_test_domain/insite_com/qs2_pwd_insite_basic.asp?ST=New&PC='"$transfer_from_pcid"'&TP='"$transfer_to_pcid"'&VT=7.4"')
basic_password=$(ssh rsync@some_test_domain "curl --silent \"http://some_test_domain/insite_com/qs2_pwd_insite_basic.asp?ST=New&PC=$transfer_from_pcid&TP=$transfer_to_pcid&VT=7.4\"")
(I hope I got that right
Note also that $(..) is preferred over `..`
PS: Please use [code][/code] tags around your code, to help preserve formatting and to improve readability.