What guidebook says women walking alone in Montmartre in the evening will be harassed or uncomfortable? I think they were probably meaning as a woman when you are walking alone after a certain hour, you should just try to remain alert and aware of your surroundings - this is going to be the case for any area of Paris (or any other big city around the world)
Yes, there are a couple of "one-offs" in Montmartre that you should actively seek to avoid or at least keep your guard up (and this applies during the day - not just at night). In a tiny handful of streets immediately near one of the Metro stops there is a stretch of stores that sell souvenirs to tourists. If you accidentally find yourself walking down those particular blocks, expect to say "No" to a small minority of pushy salespeople - and keep walking. Near the base of the funicular to Sacre Couer, you may encounter a few of the friendship bracelet guys. They are very easy to spot, but some of them are pretty brazen and don't fully understand a very firm NO. Even though I never once broke stride, I had one that followed along side of me and tried to grab my wrist until I really ripped into him and then he slunk off to the next tourists. Don't be the genteel southern woman with these guys - they aren't trying to physically hurt you, but they are trying to get you to stand still for a minute so they can get a bracelet on you and then try to demand payment.
On the steps of Sacre Couer and over near the square where the artists hang around selling portraits and paintings to tourists, you will also find a handful of similar "actors" - petitions, deaf/mute girls, etc. Again, you will see them coming. There is nothing they hate more than someone who is very vocal and who draws attention to what they are up to. So if you do this, they just move on.
Some parts of Montmartre are adjacent to Pigalle. I don't find Pigalle to be particularly scary, but some people absolutely hate it as there are some pockets where you'll encounter strip clubs, a couple porn shops, etc. If those things make you uncomfortable, avoid staying somewhere that borders Pigalle.
That said, there are many other sections of Montmartre that are perfectly lovely. When you start walking away from the tourist draws, you will find parts where there are just a lot of nice restaurants, cafes and plenty of local businesses. Some sections are absolutely beautiful and very peaceful. But without knowing exactly where this apartment is that you are looking at, it's hard to say if you are in one of those sections, or if you are right near tourist central.
While it can be a very pretty area, it can also be quite hilly around there (think San Francisco and make sure you have a decent fitness level when it comes to walking a lot). If you focus on the arrondissements that are in the double digits, you will likely get more space for your Euro in terms of apartment rentals - but remember that some of your savings will be offset by the amount of money (and time) you are going to spend in transit to get to the tourist sites in the single digit neighborhoods.
I really like parts of the 10th and find it great to stay as I get more space for the money and many areas are full of both white and blue collar locals just going about their lives, so the stores and restaurants seems to keep some of their prices lower. It's not everyone's cup of tea - some people get very nervous if they have to use one of the big train stations like Gare du Nord or Gare de l'Est in the 10th.
Bottom line, stay wherever you feel it makes the most sense for you from your budget and preferences.