@Katie1023 I get these inquiries as well, and I always turn them down because I don't have an Instagrammy place, plus I'm pretty much always booked so no need for further promotion. I also think that most of the inquirers are chancers. However, if you're interested, there are a few questions you should ask yourself and the influencer.
1) How many followers do they have? Many people seem to believe they are influencers, but they have maybe 10k followers. That isn't an influencer. If they have 100k followers (or are in some niche market) maybe it's worthwhile, but ask that question.
2) What do you hope to get out of it? Just the promotion through IG, or do you want nice pics to add to your listing? If the latter, make sure you spell that out really clearly in your agreement. If you have something else you're expecting from them, again, spell it out really clearly.
3) Do you really feel it will benefit your listing, and if so, how specifically? Make sure you know what your end game is.
This likely won't be your last inquiry of this type, so it's good to work these things out ahead of time. Remember that most of these people are spamming multiple hosts with the same request.