Kathy, Rione Monti is just the official, full name of the district. Without an address, we still can't tell you exactly where you'll be although from the description I'm sure it's on the north side of it. It's fine; you don't need to make a different choice. There is no place in Rome so central that you could avoid taking transit altogether if you're not up to going everywhere on foot. We haven't found that to be difficult but we have hopped the metro/commuter train now and again when going farther afield than you probably will, or wishing to cover a sizable stretch of ground more quickly.
We have never taken a taxi.
Pickpockets; yes, Rome has them. So does any other large tourist city. They are simply to be outwitted rather than feared, and that is not difficult to do. A lot of RS posters swear by moneybelts and neck wallets. Others have hidden, interior pockets. Our method of choice is a Pacsafe day bag with locking zippers (Travelon makes them as well) and/or I pin a small zippered pouch inside my waistband for my daily cash: virtually invisible and easy for me to access but no one one else.
Whatever method you choose, the trick is to disperse the location of your valuables so they're not all in one place, and make it difficult-to-impossible for light fingers to access them without your knowledge. Never carry valuables in a purse or put anything you don't want to lose in an exterior pocket. Don't even use an interior pocket unless it's one designed to foil a thief. Once we've checked into a hotel, we keep some of our cards and bulk of the cash in the room safe and take out only what we need for the day.
Really, there's no need at all to be paranoid about the thieves. The vast majority of tourists who've been relieved of their valuables made themselves easy victims. If you just take some precautions and avoid distractions, you're unlikely to have a problem. But safety tips might be another topic for another thread. :O)