According to Merriam-Webster the definition of "criminal" is:
1: relating to, involving, or being a crime <criminal neglect>
2: relating to crime or to the prosecution of suspects in a crime <criminal statistics> <brought criminal action>
3: guilty of crime; also : of or befitting a criminal <a criminal mind>
Since DUI is certainly a crime, and "another alcohol related offense" implies further criminal activity, your father certainly IS a criminal. Like others here, I do not mean to be callous or judgmental. But, to effectively answer any questions you may have, a realistic view of the circumstances in necessary.
As already noted, the qualification criteria for having publicly funded legal defense is primarily based on visible income. The criteria is "if you cannot afford an attorney," not "if you choose to spend your money on other things." Your father may well have to change some spending decisions to pay for legal counsel. I know that to you this may seem like an unfair hardship, but it really isn't.
Without knowing what charges your father is facing, especially the other "alcohol related offense," along with some further specifics, nobody can even tell you what the likelyhood of jail is. He is not likely to face any significant jail time, even if convicted, for a first-offense DUI. If the other offense is something like a drunken disorderly conduct, that is also not likely to result in anything more than an over-night jail stay. If it is a drunken chainsaw murder, on the other hand…well, that could well be a different story any you may be right to worry.

