There are RER lines and stations throughout the city, not just along the left bank of the Seine. The RER B line runs from CDG airport south through the city and continues to southern suburbs. I've taken it dozens of times. In order to get to the Hotel Brighton (which is across from the Tuileries on the right bank and along Metro line 1, one would transfer from the RER to Metro line 1 at the Chatelet station (which is on the Right Bank). That transfer is usually quite long, but there are moving walkways for the longer transfers. Chatelet is probably the busiest RER station in the city, and the Metro station may also be the busiest because so many different lines go through it.The RER trains stations are along the left bank of the Seine. Such as the station under the Musee d'Orsay or St. Michel in the Latin Quarter.
From there, you would probably have to switch to a subway line and then walk some distance with luggage to the hotel. It could be close or it could be a couple of blocks.
With reasonable precautions, I think both the RER and the Metro system are pretty safe. I've taken literally hundreds of trips on the Metro over the past 35+ years, including last year. In all that time, there was one unsuccessful attempt by gypsy kids to steal from me. I was seated on the metro and I shoved the kids away forcefully the moment they attempted to distract me by waving a crumpled piece of newspaper in front of me. Immediately after I shoved them away, the Parisian man sitting next to me did the same. The kids got nothing from either of us. I haven't seen gypsy kids in Paris for about 10 years. I hope that the authorities finally cracked down on them. BTW, you get a free transfer from the RER to the Metro. You typically need your RER ticket to enter AND exit from the RER, and you'll use the same ticket to enter the Metro system. You don't need the ticket to exit the Metro, however.
Map of the Metro & RER network: http://www.ratp.fr/plan-interactif/
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