Staying in the 12e versus 9e in Paris - solo female traveller

Safety is really not an issue in Paris and not in either of these locations. There is nothing unsafe about Pigalle although it is unlovely. You can confidently travel the metro at all hours and find it full of couples, groups of women, groups of men, singles etc. I have literally traveled from one end (Alesia) to the other end (Malesherbes) at midnight without being nervous. The streets of Paris tend to be full of people till quite late and there are sidewalk cafes with people eating and drinking till late.

I would google map walk the area around each apartment and see what the immediate environs are. Either of these is fine. The 9th will be more touristy.

I second what Janet said about safety; there is very little violent crime in Paris; pickpockets will probably be your biggest concern. As to where to stay, I am going to give the nod to the 9th simply because it is more centrally located than the 12th. I once lived on the Rue des Martyrs not far from Mo. St. Georges and it makes for a nice walk up a bustling street that is very untouristy and very Parisian. Rue des Martyrs starts near Notre-Dame de Lorrette and ends near Mo. Abbesses. You can then walk over to the funiculaire that will take you up to Sacre Coeur and Montmartre if you wanted. Enjoy your trip!

I have twice stayed in apartments in the 9th and found it very attractive. Lots of restaurants & shops, of a very trendy variety, to suit the media & arts Parisians who live there. I like the 12th as well and it has the added convenience of being reached from CDG by Bus No. 351 directly to Nation for the princely price of 3 bus tickets, which you can buy from the driver. Incidentally, Bus No. 350 will take you from CDG to Gare de l'Est, from where an inexpensive taxi ride will get you to your hotel in the 9th. To address the safety issue: there is no safety issue.

I'm writing this from the 9th, and I assure you that the area between St. George and Pigalle is very nice. We've been here three years in a row and watched it become trendier each year. You would go up the hill toward Pigalle, which isn't so bad anyway, only to get to Montmartre. Everything else is down the hill toward St. George and beyond.
Kim lives in the 12th, so maybe she'll chime in.

Either one would be fine, but I would vote for the 12th near Picpus/Nation. This is exactly the area where I was recommending a couple of places to Nigel a couple or three months ago, maybe you can find that thread by googling. Honestly with the metro at Nation you can zip "into town" (I.e. the center) via the line 1, and you also have the lines 2, 6, and 9 as well as the RERA. If it's one of the hotels right at Picpus, there are lovely huge elegant buildings right there at Square Courteline. If you want to PM me the name of the hotel, I probably will know it. This area is upscale and safe (a lot of retires live here, plus young families). saint Georges is a little more hipster, perhaps, but not any less safe either.

If it's the Hotel Lux, it looked fine when I went and checked out the lobby one time, and there are all sorts of wonderful food shops around the corner on the rue de Rendez-vous.

I have felt comfortable wandering both areas alone. At least until the metro closes, you should find other people around. See also some tips for traveling alone at http://www.ricksteves.com/travel-tips/trip-planning and some encouragement to wear your moneybelt at http://www.ricksteves.com/travel-tips/theft-scams.