The issue is choosing between à or de.

When you say Elle refuse de boire you have the verb refuser followed by another verb, de is the proposition you have to use after refuser when it is followed by another verb (to refuse to do something).

When you say Elle refuse à boire you still have the verb refuser, but in that case it is followed by the noun phrase à boire. À boire means “a drink / something to drink”. It is the same construction as if you said: Elle refuse le cadeau. In those two sentences à boire or le cadeau are direct objects of the verb refuser.

No, there is no rule for which preposition follows a verb in cases like this. You have to look it up in a dictionary to be sure.

refuser de faire quelque chose

continuer à faire quelque chose

commencer à faire quelque chose

essayer de faire quelque chose

Some verbs can even take different prepositions in different cases. Sometimes there appears to be some logic about this:

penser à qqch (=think about something) / penser qqch de qqch (=have an opinion of something)

Other times it’s hard to discern any method to the madness:

jouer à (un sport) / jouer de (un instrument)

I don’t think there are any rules that account for even that subset, let alone all the other verbs that take either à or de. So you do have to consult a dictionary and memorize them.