The Enterprise (original) and Enterprise-D both have their exploratory missions which form the basis of the shows. Presumably not all Federation starships are involved in explanatory missions at all times. So my question is how many Federation starships during (1) the TOS-era and (2) TNG-era (each) were on exploratory missions?
The primary exploration ship during the time of Star Trek: TOS was the Constitution-class cruiser of which the USS Enterprise was the most famous ship of that line. There were 12 such ships engaged in the second exploratory phase of the Federation as it began to expand and meet other alien species. This expansion would eventually be limited by the Romulan Neutral Zone and the border to Klingon Space.
In 2267, there were approximately twelve Constitution-class starships in the fleet. (TOS: "Tomorrow is Yesterday") These included the NCC-1700, the USS Constellation, the USS Defiant, the USS Enterprise, the USS Excalibur, the USS Exeter, the USS Hood, the USS Intrepid, USS Lexington, and the USS Potemkin.
Despite the successes of the class, exemplified by the performance of Kirk's ship, the mission parameters for the Constitution-class also meant that the vessels of the class operated under highly dangerous circumstances, resulting in a relatively high loss rate, and that being assigned to one was hazardous at best. REF: Memory Alpha > Constitution-class starships
- The Constellation, the Intrepid and the Exeter were lost during their voyages. The Excaliber was crippled during war games with the supercomputer M5. The Defiant was lost in Tholian space and eventually traveled to the mirror universe.
By the time of the Next Generation, the Federation was far better equipped than in their first expansion phases and had thousands of smaller exploration ships available for duty.
Of the Galaxy-class starship, at the time of TNG, it was the most powerful starship in the Federation Fleet. The Star Fleet Technical manual said there were only six when we first see them in The Next Generation.
By the time of the Dominion War, the Galaxy Class are seen in almost every battle particularly during the Dominion war where they were on the front lines. There are at least ten shown during the Dominion War.
Analysis of the battle from "Sacrifice of Angels" showed no less than ten Galaxy-class vessels in the combined Federation fleet during Operation Return. This would indicate that there were more than the initial six Galaxy-class starships referenced in the Star Trek: The Next Generation Technical Manual.
The total of six ships stemmed from Roddenberry's assumption that Starfleet would not have constructed vessels of that large a size in substantial numbers. The Manual however also stated that an additional six space frames were constructed and laid up to be completed at a moment when circumstances so dictated.
It can be conjectured that those circumstances arose after the first Borg incursion and that by the time the Dominion War began, all the remaining original Galaxies were operational. It could be theorized that a "Galaxy-wing" would be much like modern battle groups with lighter frigates/ships screening the main capital ship(s) (which would make sense due to the size and firepower of a Galaxy-class starship). REF: Memory Alpha > Galaxy-class starship