The Borg are back...
After the Borg encounter in First Contact, Starfleet would almost certainly have reacted by sending out vessels to protect outer colonies, especially those close to the Borg cube's flight path. Even if Starfleet Command were to conclude that the Borg would probably not send another cube for some time (based on the time elapsed between "Best of Both Worlds" and First Contact), it would still be unwise not to beef up its presence out at the fringes.
There is a political element at work here, too. If you were living in an outer colony of the Federation and a Borg cube just passed within a few parsecs of your world, you can imagine the panic that might ensue around you, and that panic could easily last for months or longer. Your elected officials would likely press the Federation for greater protection. The Federation Council would likely respond by sending starships out to these areas.
Couple all of this with the fact that some starships were lost in the battle with the cube.
The Klingons are not to be trifled with...
By the time of TNG, the lengthy border between the Federation and the Klingon Empire no longer features the defence infrastructure it would have in the TOS era. After the Khitomer Conference of 2293, the Neutral Zone with the Klingon Empire was scrapped (leaving only the Romulan Neutral Zone), and as a gesture of peace, the starbases and defence installations along the former Klingon Neutral Zone were "mothballed" (to borrow Admiral Cartwright's wording from The Undiscovered Country).
All of sudden in DS9, the Federation finds itself at war with the Klingons again, after nearly a century. How will they defend the border? Almost certainly by sending an armada of ships to patrol the former Neutral Zone, in order to head off potential Klingon incursions and as a sign of strength.
Those sneaky Romulans...
The Romulans have a tendency to sit along the sidelines and take advantage of situations when the time is ripe. The Romulan Star Empire's two greatest foes are the Klingon Empire and the Federation — the Romulans would be pleased as punch that they are warring against each other once again. If past actions indicate anything, the Romulans might try to stir up further trouble. They might wish to ensure that the Klingon-Federation war is as long and bloody as possible, and could try to engineer incidents that would force one side to step up their offensive against the other (such duplicity on their part is certainly not unheard of and very much in the Romulan "style").
In any event, it would have been prudent for Starfleet to heighten its presence along the Romulan Neutral Zone. That means more ships diverted away from core areas of the Federation.
There are several good reasons why, at that particular time in DS9, Starfleet would be spread quite thin.