Guessing he'd probably be out after four terms; i.e., 1983. There might well be some guilt by association, or comments like "he was simply lucky he didn't get caught", which in turn may well have led to Millicent Fenwick becoming a senator. She might have paved the way for Tom Kean to follow his gubernatorial terms with tenure in the Senate.
I for one would certainly hope that in this scenario Lautenberg only saw the inside of the Senate chamber as a visitor. IOTL, he rarely if ever hinted that he knew that New Jersey went south of a line from Trenton to Toms River, with the exceptions of Cherry Hill and Atlantic City. As far as southern NJ was concerned, Lautenberg was a consummate northern NJ hack politician who (a) rarely if ever had an original thought; (b) voted a straight party line, and (c) did nothing for the southern portion of the state. I say that as a resident of southern NJ since 1980 and in theory one of Lautenberg's constituents, so I've seen what Lautenberg and his ilk did--or rather, didn't do--for southern NJ / the state as a whole.