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As an example, the key of G is played alot due to the guitar’s default tuning of EADGBE. In the context of fingerstyle/chord-melody guitar it’s nice to be able to drone on open strings that are in the key.
So whenever I play a song in a certain key, esp. for fingerstyle guitar, should I make sure my open strings are tuned to that key as well? (and use a capo and/or change tuning of strings if needed). For example, let’s say I wanted to play a song in the key of C# major.
It is clear that some keys will have many open string notes in them. You will find that much classical guitar music is written in such keys. This is due to the desire to have open string resonances contribute to the volume of the instrument. However on the electric this is not as much of an issue. In fact on an electric open strings can generate unwanted feedback. So the motive for what to play is different in each case. You had mentioned a motive of having open strings to drone on. Yes when arranging a chord melody using open strings can (1) make the chord melody easier to play and (2) allow for some beautiful effects. Kenny Burrell's chord melodies I find quite beautiful and he will often tap out a rhythm on a low string while playing melody lines or chord melodies on the top strings (but that doesn't mean he's using open string keys all the time, he's just really good).
I am not sure what specific thing motivates your question, but I'd rewrite the tune in an open string key in certain cases. A lot of Jazz standards in the Real Book written by horn players are in Eb or Bb since those are the keys the horns read in. But there is no reason you need to follow that. Take Anthropology by Charlie Parker (in Bb) and arrange it for guitar in C. If you are playing with a group it's no problem. I play all of Parker's tunes in the Bb Real Book key and it doesn't bother me. But then again I'm not playing chord melody. As for that, you do not need to change key to play a beautiful chord melody. You can use the movable chord forms and not worry about open strings. However, some choices will be out of reach. If you really want that open drone available to you then I think you really do want to transpose to another key.
If your are arranging tunes to be played on a classical then I would be more adamant about doing it in an open string key to maximize the resonance opportunities provided by the instrument. One final point, you can create those resonance opportunities by fingering all six strings even when you are only playing a few notes. For example, the E string will not likely vibrate much when you play Bb. But if you use a full barre and grab the F on the E string then yes, it will resonate. this is because you've essentially tuned that string to F. I use this trick all the time to get more and better resonances. I will finger the whole chord even if the sheet music doesn't ask for it. This, of course, requires more skill and will reduce mobility. It's fine for some classical pieces that are not too quick but would be harder on fast jazz tunes. This is the same thing as using the index finger as a capo. So if you have a reason to stick to a key that does not have any open strings you may want to use the capo to help with this. But I am only addressing this from the point of view of playing classical guitar where the resonance is critical to a good full sound. I would never (and have never) used a capo for this. If your interest is in being able to grad an easy open string bass line then yes, either transpose the song to an open string compatible key or use the capo.