The Fifth Amendment always protects someone from being forced to testify against themselves if it would implicate them in a crime (see, among others, Ohio v. Reiner, 532 U.S. 17). Any person can assert the privilege, regardless of their role in the trial, with the possible exception of the plaintiff (who is the one person who wanted to go to court). Like always with the Fifth Amendment, they can answer some questions but not others (but if they do answer a question, they need to fully answer it).
In civil cases, the Fifth Amendment itself does not keep the jury from making adverse inferences against whoever invoked the privilege; if you refuse to testify, they can assume that it's because testifying would be extremely damaging in that particular case. However, most states have rules against that, and so invoking the privilege in state courts generally works like it does in a criminal case (where the jury basically ignores that the question was even asked). In federal courts, if a case is being heard under diversity jurisdiction (plaintiff and defendant are from different states but the claim is not a federal claim) the state rule is supposed to apply; if the claim is a federal claim, the federal rule applies and adverse inferences are allowed.
While the Fifth Amendment can be invoked by anyone, there may be consequences. In many states (where adverse inference isn't allowed), a witness who will just invoke the Fifth and answer no questions can't be called, because it's a complete waste of time. If the plaintiff invokes the Fifth to not answer key questions, then the court can potentially dismiss the case; they have the right to assert the privilege, but their lawsuit might suffer for it. In federal court, another possibility that's been done several times before is that the civil case is just put on hold until the criminal matter is resolved.
“The Fifth Amendment Can & Will Be Used Against You In a (Federal) Court of Law”
Taking the 5th: How to pierce the testimonial shield
Plaintiff as Deponent: Invoking the Fifth Amendment