Does Griffin v. California, 380 U.S. 609 (1965) not extend to civil proceedings?
No, it does not. The considerations when invoking the Fifth Amendment in a civil case are different than in a criminal case because there the party is facing the power of the state in trying to convict him/her on that offense. In a criminal trial instructing the jury that it may make negative inferences from the invocation of his right to remain silent would undercut the right and make it easier for the state to convict him/her.
But in a civil proceeding, a party could use the Fifth Amendment to game his civil case, allowing him to avoid having to to testify to things that would adversely affect the civil proceeding. As the Fifth Amendment only prevents use of the testimony in a criminal case, to allow a party to use it in a civil case where the right is not intended to apply is a problem. Still, forcing him to testify against himself would allow those statements to be used against him later in a criminal proceeding. So a balance is struck in the case law. He may still invoke the Fifth Amendment right, but unlike in a criminal proceeding a jury may be instructed that the jury may (but is not required to) infer that the answers would adverse to him. The U.S. Court of Appeals for the Eighth Circuit explained it this way:
Although the privilege applies in both civil and criminal proceedings, the Supreme Court has viewed the assertion of the privilege differently depending on whether a civil or criminal proceeding is involved. In criminal proceedings, the prosecution is prohibited from commenting on the accused's silence, and the trial court is forbidden from instructing the jury that the defendant's invocation of the privilege may be used in the jury's consideration of the guilt or innocence of the defendant. Griffin v. California, 380 U.S. 609, 615, 85 S.Ct. 1229, 1233, 14 L.Ed.2d 106 (1965). By contrast, the Supreme Court in Baxter v. Palmigiano, 425 U.S. 308, 318, 96 S.Ct. 1551, 1557, 47 L.Ed.2d 810 (1976), stated “that the Fifth Amendment does not forbid adverse inferences against parties to civil actions when they refuse to testify in response to probative evidence offered against them.”
Cerro Gordo Charity v. Fireman's Fund Am. Life Ins. Co., 819 F.2d 1471, 1480 (8th Cir. 1987)
And that same principle still applies today. This is from the most recent case I could find, issued just over two months ago:
As to the claim of error, the district court advised the jurors that they “may, but are not required to, infer” from Greer's invocation of the privilege that his “answer would have been adverse to [his] interest.” D. Ct. Doc. 153 at 15. This instruction was an accurate statement of the law and was not materially different from the adverse inference instruction we approved in Brink's.
Mirlis v. Greer, 952 F.3d 36, 44–45 (2d Cir. 2020). There are some limits to this, however, as the same court explained in 2017:
The district court gave the following instruction as part of its final charge to the jury:
[F]rom the plaintiff’s invocation of the Fifth Amendment, you may draw certain conclusions but are not required to do so. Specifically, you may infer that the plaintiff’s answers at her deposition, if she had not refused to answer, would have been “yes” to the questions asked, if she had not invoked the Fifth Amendment. You may, but are not required to, draw these inferences against the plaintiff when you are evaluating her credibility, and you can give these inferences whatever weight that you wish or, if you choose to give it no weight, you can do that.
J. App’x 642–43. The instruction accurately states the law insofar as “the Fifth Amendment does not forbid adverse inferences against parties to civil actions when they refuse to testify in response to probative evidence offered against them.” Baxter v. Palmigiano, 425 U.S. 308, 318, 96 S.Ct. 1551, 47 L.Ed.2d 810 (1976). Such adverse inferences are appropriately admitted, however, only if they are relevant, reliable, and not unduly prejudicial. See Brink’s Inc. v. City of New York, 717 F.2d 700, 710 (2d Cir. 1983).
Woods v. START Treatment & Recovery Centers, Inc., 864 F.3d 158, 170 (2d Cir. 2017). That last sentence is a reflection of the general limits to admissible evidence found in the Federal Rules of Procedure.