TAS is closest in spirit as already mentioned, and I also agree that the first season of TNG is closest in spirt to TOS.
DS9 had basically no input from the people who made TOS. (and yes, I
do think that is very relevant) TOS, while at times dealing with some fairly complex ideas and issues, almost always had the sense that in the end, everything and everyone was going to be o.k. (the only instance I can think of off the top of my head where this is not true was "City on the Edge of Forever" where Kirk is clearly wounded at the end) DS9 was a fairly bleak show, particularly towards the final few years, and had many shows where things weren't necessarily peachy-keen by the end credits (or somewhere in between a six-parter

)
I'm currently viewing through DS9 and most of the season 5 episodes I've watched over the last few days are about war, battles (hand-to-hand or ship-to-ship), loss of human life, war, deceit, treachery, abduction, terrorism, ect ect. So I don't really agree that "in spirit" DS9 is closest to TOS. TOS generally returned to the status quo, complete with a little cute exchange between the big three before the end credits. The "spirit" of TOS was indeed that humanity had yet to deal with
all of their issues, but as far as making social commentary, they typically stuck that to the alien/alien race/planet of the week -
just like TNG almost always did. We didn't hear much of anything about social issues or other problems on Earth during TOS. Spock would make commentary about humans, but when he wasn't making comments about Kirk or McCoy's (usually harmless) human emotions he was usually talking about humanity in a
past tense. (a comment that usually had something relevant to Earth in the 1960's)
While many still like to make comments about TNG's "perfect" characters, I still maintain the TNG season 1 and 2 are in so many ways a different show than TNG season 3 through 7.