Karen , there are many nice areas ( arrondissmonts) to stay in .
I like the 6th ( St Germain ) , the 5th( Latin Quarter) , the 4th ( Marais) are also nice enough.
When I was on a RS tour this summer , we did not stay in the 7th( Rue Cler area) , we stayed in the 14th which was a first for me( I usaully like 4, 5 , and 6 best) , but was a great area also .. I don't think you there is any one best area frankly.
It really is helpful to check out the transport options closest to where you are staying, it is super convenient to be near a metro and RER ,, although the RER is not as important.
Also, if you are the type who wants to walk everywhere and not use metro or buses much, then I think it is best to stay closer to the river, and nearer to Notre Dame, between Musee Orsay on Left Bank and Boulevard St Michel which traverses Latin Quarter and ST Germain .
I have never heard about avoiding the 10th, thats news to me.
I was told there are some good budget hotels in the 13th, I was going to stay there a few years ago with a friend who is Chinese, she said she had friends in China Town we could stay with , and that it was in the 13th, but I can't verify that though as she ended up not coming and I ended up staying somewhere else. I did look up a few places on tripadvisor.com in the 13th though and they looked decent and area seemed to have good reveiws( I didn't want to stay with her friend , but I agreed to stay near her friend).