This question already has an answer here:
- “Would you mind to do something?” 1 answer
- Would you mind arranging to travel with us
- Would you mind to arrange travelling with us
The stem Would you mind takes a gerund; furthermore, the verb arrange, when followed by a verbal, takes an infinitive. So as far as grammaticalness goes, the first sentence is the correct one. (If you're looking for a difference in meaning, I don't think there is any: The second one would probably be interpreted the same way, even if it is nonstandard.)
If the meaning is supposed to be
then the first sentence sounds maybe a little too indirect/polite to me. I guess it's the contrast of "Hey, why don't you travel with us? We can sit together on the train and eat together and everything!" and "I'm going to be really indirect and subjunctive and stuff when I'm asking".