Is it legal to set up on-line banking for my father who is medically unable to pay his bills, so that I can pay them?

There is nothing improper from a legal standpoint for you to establish an online banking account for your father. However, I suggest that if he is mentally competent that a power of attorney be created for him naming you as his attorney in fact (assuming one is not already in place). The power of attorney should be a broad power allowing you authority over business and medical matters.

I suggest that a Wills and trust attorney be consulted with concerning drafting such a document as well as a Will or trust if need be.