Is it permissible to benefit from a program purchased online, with fees paid to a non-Islamic bank?

What is mentioned in the question is not sufficient to envision the purchase process and rule on it. As for the bank taking fees, it falls under the permissible categories of brokerage (samsara) or compensation (ju'ala), subject to their conditions.

It is not permissible to deal with interest-based banks except in cases of dire necessity and when no Sharia-compliant alternative is available.

Since the purchase of the program has already occurred, there is no harm in you keeping and using it, as the prohibition is not related to the program itself.