The first, and arguably most important service is that they should have a shuttle. Most long term parking areas are not right on the airport property. Instead, they are privately owned lots that may be across the street or down the road from the terminals. In these cases, you should be able to park and then catch a bus or shuttle to the terminal of your choice.
Also, make sure these shuttles leave frequently. They should run at least every 30 minutes, if not more often. If they leave only once an hour, that could actually lead you to being late for your flight. They should also be available 24 hours a day, 7 days a week for red eye travelers.
Most of these lots have fences around them, but fences alone are not enough to deter possible car thieves. That's why you should only go with a company that has additional security measures to protect your vehicle. A car is far too expensive to leave up to chance. Some security measures that are good to look for include cameras with remote monitoring and recording. Perhaps they might have a security guard around the clock to deter fence jumpers as well.
If you live in a very hot or sunny climate, you may want to rent a spot from a place that has covered parking. Imagine coming home after a long trip to find the inside of your car so hot that you can barely sit on your seat. Or perhaps there was a freak hail storm while you were gone, and now your car is damaged. Avoid these possible issues by going for covered parking, even if it costs a little bit extra.
If you really want to get luxurious and full service when it comes to amenities, you can look for one of the companies that offers car washing service. If you like having a clean and tidy car, and hate getting into one that is dirty, there are some that offer full washes and even interior cleaning such as vacuuming and air fresheners.
If you travel frequently for business or pleasure, ask about discounts. Even with the fancier amenities, if you park frequently, some will still offer you big discounts. Ask about a loyalty club or bulk discounts and see how much you can save.
Of course, it is not a good idea to focus only on price. Sometimes it is worth it to spend a little bit extra on your long term international airport parking to get that covered spot, or go with a parking lot that has extra security. Do what you feel comfortable with that is within your budget, and have safe, worry free travels.