Currently the Senate is in Republican hands, with a margin of 54 to 46 (technically, there are two Independents, although both of those caucus with Democrats: Bernie Sanders & Angus King). However, that lead will almost certainly shrink after the November election. Regardless of any purported "drag" effect from Trump, Republicans in this election are defending 24 seats, while Democrats are defending just 10. Historically, it's harder to defend this many seats without losing more than you gain.
This year, Democrats merely have to take 5 seats from Republicans, while holding onto all of their current seats, to gain a slim majority, and so far they have a good chance of doing just that, according to Larry Sabato, political scientist at University of Virginia, who runs the UVA's Center for Politics. His analysis shows that all of the Democrat seats are safe, except for Nevada, where Senate Minority Leader, Harry Reid, is retiring, and that race is currently too close to call. In contrast, of the 24 Republican Senators up for re-election, Sabato says that 2 are "likely" to lose to Democrats (Wisconsin & Illinois), while 3 are "leaning" Democratic (Indiana, New Hampshire & Pennsylvania). While the Indiana race would not have originally been very competitive, with the entry into the race of Evan Bayh, a popular former Indiana Governor, the race is now polling significantly in his favor--the most recent polls have him up an average of 17% over his challenger.
In fact, depending on who wins the presidency, and it strongly looks like it will be Clinton, the Democrats really just have to win 4 seats, since that would give them a tie, and Vice President Kaine would presumably break any ties in favor of Democrats. That means that even if the Nevada Senate seat goes to a Republican, but the rest of the seats go in the direction of current polling, Democrats will technically control the Senate. In any case Democratic hold over the Senate would be tenuous, with either a 50-50 tie, or 51-49 as the most likely scenario.
The top section of the table is where Sabato calls the seats "leans," plus the one toss-up, Nevada. The polling in yellow are results that were obtained prior to those state's primaries (designated in the middle column labelled "Prim")--in this case, just Arizona & Florida. While McCain's (AZ) early lead seemed quite large (13%), the only poll since the August 30th primary shows that race is currently tied.
The "lighter" red and blue are polling results that are between 3-5% for Republicans or Democrats, respectively, which I would consider "weak" leads, if at all, since these are likely within the margin of error. The green poll results are within 0-2%, or basically just a tie. My assessment of the polling tends to match Sabato's.
Current state-level polling, and the UVA's politics site both seem to indicate that Democrats have a good chance of taking the Senate in November, either with a tie, or at most, a 51-49 lead. What seems less likely, is the coincident situation that Reid's seat remains in Democrat hands, AND McCain's seat also falls to Democrats. But even then, the Democrat win margin would only be 52-48, a long way from a filibuster-proof majority. And either way, there are few analyses that are predicting a Democratic win for the House, undoubtedly leading to a bitter 2 years (if not 4) of Democratic & Republican wrangling for control of the federal budget & political system.