DS9 "In Purgatory's Shadow (Part 1)":
SISKO: A Dominion invasion of the Alpha Quadrant will affect Cardassia every bit as much as it's going to affect us. Besides, we need all the help we can get. The Dominion picked a perfect time to invade. The Cardassian fleet is in shambles, the Romulans are not much better off, and between the Klingon War and the recent Borg attack, Starfleet's spread pretty thin.
This episode takes place in 2373, the same year as the movie First Contact.
Star Trek: TNG movie First Contact:
FLEET COMMUNICATIONS: Flagship to Endeavor. Standby to engage at grid A-fifteen. ...Defiant and Bozeman, fall back to mobile position one. ...Acknowledge. ...We have it in visual range. A Borg cube on course zero point two one five, speed warp point nine six.
BORG COMMUNICATIONS: We are the Borg. Lower your shields and surrender your ships. We will add your biological and technological distinctiveness to our own. Your culture will adapt to service us. Resistance is futile.
FLEET COMMUNICATIONS: All units open fire. ...They've broken through the defence perimeter. ... Continue to attack. ...We need reinforcements. ...Ninety-six dead and twenty-two wounded on the Lexington.
Hardly a disaster for the fleet.
(the Borg cube, under attack, approaches Earth. The Defiant is one of the attacking ships) [Defiant bridge]
CONN OFFICER: Main power is off-line. We've lost shields and our weapons have gone.
WORF: Perhaps today is a good day to die. Prepare for ramming speed!
Again it's one ship, and the Defiant is happy and healthy over at DS9 for this episode, so it didn't affect the fleet.
CONN OFFICER: Sir, there's another starship coming in. ...It's the Enterprise!
RIKER: The Defiant's losing life support.
PICARD: Bridge to transporter room three. Beam the Defiant survivors aboard.
So by this episode whatever damage was done to the Defiant had already been repaired.
PICARD: They're creating a temporal vortex.
RIKER: Time travel!
(the Enterprise suffers a shockwave)
This is the key point though, the Borg goes back in time.
PICARD: Data, report!
DATA: We appear to be caught in a temporal wake.
WORF: Captain, ...Earth.
DATA: The atmosphere contains high concentrations of methane, carbon monoxide and fluorine.
PICARD: Life signs?
DATA: Population ...approximately nine billion. ...All Borg!
PICARD: They must have done it in the past. ...They went back and assimilated Earth. ...Changed history.
CRUSHER: Then if they changed history why are we still here?
DATA: The temporal wake must somehow have protected us from the changes in the time-line. (Data's console bleeps)
DATA: Sir, the vortex is collapsing.
PICARD: Hold your course, Mr. Hawk. We must follow them back. Repair whatever damage they've done.
So at this point in the movie in 2373 the Federation now never was created, in fact the only humans we know that still exist outside of Borg influence are the crew of the Enterprise. The only Federation ship that exists is the Enterprise at this point in the movie.
The movie ends with them returning to their correct time.
DATA: Helm standing by.
PICARD: Mr. Data, lay in a course for the twenty-fourth century. I suspect our future is there waiting for us.
DATA (OC): Course laid in, sir.
So Picard acknowledges that things are back how they were.
Once Picard returns I'm not sure exactly how much of everything the Federation is going to know about.
It doesn't appear that this was a major confrontation that would weaken Starfleet in any way. They mention a couple ships that took damage. The Defiant was repaired by the time of the episode "In Purgatory's Shadow" so I assume the other ships were as well.
What Borg attack is Sisko talking about that spread the Federation thin? If it is the battle in "First Contact" that he's referring to, how did that attack spread the Federation thin?