I know the Senate briefly went into GOP hands 1947-48 and 1953-54, but do we have to count those? If not, I'm not sure how to answer the challenge -- the GOP wins overall following the war seem pretty hard to avert. But if losing the body for isolate two year intervals is allowed, then I think it may be possible to delay what happened in 1980 from happening until 2012 or after.
That's said, let's start by looking at when the Democrats had the biggest majorities -- the high point was undoubtedly after the 1936 Elections, when the Dems controlled 76 of the 96 Senate seats. By the start of WWII, this majority had gone down to 66 seats, and they would go down further for the next several elections... You know what, though, this seems like a little early a PoD to sustain the lead.
So let's see... Ah! This pre-war level majority would not be matched again until the 1964 Elections (interesting note, this was just about entirely due to massive gains in 1958, following a nearly tied Senate, with small additions in 62 and 64). OK, so we've got a good high point -- now OTL, there were net losses in 1966, 68, and (counter-intuitively) 70 that should be easy enough to contain -- they were 13 total losses over six years, with enough GOP pick ups being narrow enough to make it five with just a little extra luck. So TTL would see Democrats holding 61 Senate seats in 1971, instead of 54, which in turn could mostly be made up for with the small gains (again counter-intuitively) that OTL saw in 1972 (three Democratic net gains OTL, plus the OK win was narrow, so potentially a net gain of four).
Assuming TTL still has President Nixon (and I can't see why it wouldn't be able to), who still gets in hot water over Watergate (again, why not?), 1974 will be a good year for Democrats -- OTL saw a net gain of four Democratic seats in the Senate, with four GOP won races (mostly re-elections) that were really close. So by making the years 1966, 68, and 70 more just a little more stable, mostly leaving 1972 alone, and giving a little bit of gas to 74, the 1975 Democratic majority in the Senate goes from 60 out of 100 to 73 (or 74, counting Harry Byrd). That's practically 1937 levels! (And, incidentally, with these relatively small changes, the Dems still this supermajority a decade following Civil Rights... so that's containable as an issue.)
So where would the Senate go from here? Well, OTL 1976 was stable, there were losses in 78, and 1980 was enough to turn the body to the Republicans, joining the tide of the Reagan Revolution. So let's say Ford beats Carter in 76 -- Senate races there go as OTL, but now the country is in an anti-Ford mood in the late 70's, so these losses can be largely averted (plus the Dems will win the Presidency in 1980). If at least 70 of 100 Senators are Democrats in 1981, assuming the President does well enough that the country isn't turning on the party in 84, the midterms of 1982 and 86 shouldn't be enough to take away the majority.
What happens then is harder to determine...