House Targaryen only.
Challenges : Choose a participating character you control. Until the end of the challenge, that character gains melee. Then, each participating character controlled by an opponent gets -1 STR until the end of the phase.

Does the STR reduction affect characters that enter the challenge after the event is played?

I would say no. The events resolves and goes to the discard pile. Then your opponent fleets a character into the challenge as a standard player action later in the same challenge, and that character has no knowledge of dragon support being played.

But I am probably wrong.

You may want to send it in to FFG to be sure, because there are some very subtle things going on here.

It's not so much because of the timing aspect kr4ng cites (after all, events or other triggered effects can create lasting effects that change the environment, and thus could apply to different characters as they change their interaction with the environment). Rather, it is more about the exact wording changing the interpretation.

For example, if an effect read:

"Challenges: Choose a character. Until the end of the phase, characters participating in a challenge against that character get -1 STR."

You would interpret that as applying to any character that found itslef looking across the battlefield at the chosen character at any time during the phase (and the -1 STR going away as soon as they were no longer on the opposite side of the challenge). However, if an effect read:

"Challenges: Choose a character. Characters participating in a challenge against that character get -1 STR until the end of the phase."

You would interpret that as applying the -1 STR to any character currently opposing the chosen character at the time the event was played, and having that -1 STR last, not while they were participating against the chosen character, but rather until the end of the phase, no matter what else happened.

Now, look closely at those 2 abilities; they use exactly the same words; only the order has changed. So the meaning and interpretation of the effect changes significantly based on how the ability is phrased.

So, because Dragon Support says what happens to a specific set of characters until the end of the phase, it only applies to the characters that are currently in that "set" when the event is played. Compare that to something like Robb Stark ("Any Phase: Name a Trait. Until the end of the phase, each character you control that has that Trait gets +1 STR") who says that, until the end of the phase, something applies to a set of characters.

Anyway, because Dragon Support says "X happens until the end of the phase" (instead of "until the end of the phase, X happens"), I'd agree with kr4ng's result. It's a very subtle difference to something like Robb Stark, whose bonus applies to any character over the course of the phase, though, so if people don't agree with the semantics "lesson" above, they might want to send it to FFG.