Sure you could. Let's say there are 100 planets in the Federation and each one has an average of a billion people. After "First Contact", a call goes out to replace the losses that Starfleet took. If a thousandth of a percent of the citizens answer the call, you've got a million new recruits. That's more than enough to restaff the lost ships, especially since the Enterprise is the only one that ever actually does anything. Every other ship seems to be guarding the mail run between Earth and Alpha Centauri.
Like the Japanese Navy after Midway you replace crack pilots with raw recruits. Like I said, Starfleet would not have recovered for a generation.