Baroque composers wrote either specially for a soloist in their own Court orchestra, or in a range easily played by either a flute or a violin - the most common soloists. Oboe fits the range too. Extremes of range were not always used. If you disregard the high altissimo of the flute range then it has a very close written range to the saxophone, and the Bb soprano is only pitched a tone away as you know. The oboe has the same written range as the saxophone.
There are many arrangements of Baroque pieces for the sax. It has to be decided whether the piece will fit into the range - if it has a big span - and as Jbt said, where it sits best on the instrument. If the piece had a range of 2.5 octaves then you have no choice but to make low Bb your lowest note and high F your top note. If the range is considerably less then you have a choice, and you'd think about the character of the piece before choosing where it might sit best on the instrument, as well as which key you might want to play in. People talk about the "meat" of the instrument, meaning the best part of the range, or where it really sings.
Think about the tune Vocalise by Rachmaninov. This has been arranged for many instruments including sax and clarinet. It has a relatively small range so the choices are many for where you want it to sit on the instrument. By memory (I might be wrong) I think that both sax and clarinet arrangements start the tune on G just on top of the stave. It sounds melancholy there and is easily playable. Think what it might be like to play, or more importantly, to listen to if the tune started a 5th higher on top D, or even middle D. I'm not sure that I agree with Colin's synopsis of keys/feelings as it's surely dependent upon the particular tune - ie E being mysterious - parts of The Rite Of Spring are in E and I think that's more violent in most places than mysterious. Having said that - listen up Colin - Holst's Neptune the Mystic has large parts in the key of E.
Keys do sound different though when you play around with them on the same tune. Try Vocalise starting on F instead - or any other note!
Those are the important factors. There is no rule or obligation to keep it in the same key as the original. Sometimes this just won't work anyway because of the differences between instruments and what resonates well on one might not on another.