star trek - Why would any Federation starship (such as the Enterprise) have any spouses or youth onboard?

The original Star Trek series and Star Trek movies basically portrayed the Star Fleet culture aboard ship as similar to the U.S. Navy. The culture is predominantly male but women are allowed on most ships, especially in "support" roles such as science, medicine and communication. Despite a certain level of romance aboard ship through TOS, though, wives and children are not found on the original USS Enterprise, and its complement is only about 400 people as a result.

In Star Trek: the Next Generation, Deep Space 9, and the new Star Trek movie, ships of the fleet are seen carrying a substantial complement of youth and non-Starfleet types. The Enterprise-D, in fact, has a complement of over a thousand, many of those being families and children of serving Starfleet personnel (which had to be evacuated in Generations before separating the saucer). The U.S.S. Kelvin in the new Star Trek was carrying James Kirk's mother (apparently a noncombatant), who gives birth to him on a shuttlecraft as his father assumes command of the Kelvin and holds off the Romulan ship that attacked them.

Why the difference, in-universe? Starships are dangerous places, as the new Star Trek demonstrated, and even if the personnel would be in space for months or even years at a time on rotation, and even if the ship's stated mission is exploration, it would seem foolish to risk the lives of non-combatant women and children in a battle with a Romulan cruiser, or by being on the wrong end of a scientific experiment like the Soliton Wave, or encountering the wrong spatial anomaly like Tyken's rift. I understand the presence of family aboard DS9, which if nothing else is a stone's throw from a relatively safe haven on Bajor, but a ship intended "to boldly go where no one has gone before" has to be crewed with the understanding that it may not come back.

Answer

The Enterprise-D was a 'starship' and not a 'warship' (except in the episode Yesterday's Enterprise which was an alternate timeline). Its primary mission was exploration (although, it seemed to spend a lot of time in the Federation core as well) - so the analog in a historical context would be more like the expeditions undertaken by Cook and Magellan. Although not common, some expeditions in the 17th, 18th and 19th centuries carried some of the officers' (and even some of the crew's) families with them - especially when the voyage was expected to last several months or more.

Families on long-term exploration does make sense - for morale and stability.

Galaxy class starships also have the ability to separate the saucer as a 'lifeboat' while the star drive section engages in battle - and it has been speculated that most of the standard design starships (including the original Enterprise, a Constitution class) could detach the saucer - although, reconnecting many of these classes required a space dock due to explosive bolts.

As Xantec mentions, there is an observation bias with TV episodes (and movies) - we only see the most exciting events (26 episodes per season [not counting/excluding doubles] with maybe 3 days passing in your average episode [not counting time loops] only adds up to 80 days out of 365 over the year that passes in the season).

On the other hand, ships such as the Defiant, which was specifically built as a warship, have no apparent provision for family quarters (or even recreation). And it can be assumed that the larger class ships used in planned battles against the Borg and Dominion have either safely removed their civilians to a suitable planet away from the battle, or never had them aboard to start with.