If you gave the phisher your personal information, the bank will/would close your accounts.

The second sentence is correct, but it helps to understand why. There are (at least) three types of "if clauses" which set up a condition. When the "if clause" is in the simple past, then one uses "would" and the infinitive (the bank would close your account). Had the "if clause" been in the present indicative, either "will" or "would" could have been used. This is a "Type I" if clause, in which the condition (giving one's personal information to the phisher) is possible. A "Type II" if clause lays out a condition which is possible to fulfill in theory. A "Type III" if clause lays out a condition which it is not possible to fulfill. With each type of if clause, the verb form of the independent clause (the bank will/would close your account) is determined by the verb form of the dependent clause--the if clause.

I don't want to go into this too deeply, but there are also sentences in which the if clause comes at the end, and the verb forms are much the same, usually.

Thanks, Setanta, for the detailed informative reply.