As mentioned above, there are several movies in the GITS franchise, and being more specific would be helpful.
Have the Ghost In The Shell animated movies made any profit?
Not according to box office sales for those movies released in theaters.
"How can a series so good and influential flop financially?"
You are trying to correlate how a movie is received by critics and/or fans with how well it does at the box office. But, unfortunately, there are many factors that affect how well a movie does at the box office; positive reviews don't guarantee financial success. Just to recap, I put the budget and gross for the above mentioned movies based on IMDB, Box Office Mojo, and Wikipedia for regional releases.
Ghost in the Shell (1995)
- Budgeted: approx 5.4 million USD
- Domestic Gross: $515,905
- Foreign (Japan, UK only): $2.3 million USD
Ghost in the Shell 2: Innocence (2004)
Budgeted: approx 18 million USD
Domestic Gross: $1,043,896
Foreign Gross: $8,745,755
Do the box office figures include income from Japan?
In some cases, a movie may be released in different countries during different times. You may be strictly looking at a specific regional release. Your absolute best bet to look at Box Office - Worldwide, where both domestic and foreign gross totals are included. This will give you a better idea of how well the movie financially performed.
Do the box office figures include DVD sales?
No. The box office gross is basically the total of all the ticket sales for that movie.
Or something else? Surely they wouldn't keep financing something unprofitable.
The GITS franchise does exist outside of box office movies. When you consider other sources of revenue such as the manga, direct-to-dvd or blu-ray sales, merchandise sales, international licensing fees, etc. you'll start to see the bigger financial picture. It is not uncommon for manga publishers to use anime to increase revenue for their source material which is almost always the manga. In fact, if an anime release doesn't perform well, but it boosts manga sales, it's not considered a loss at all. It's the cost of doing business. How the anime industry produces profit is very fascinating because sometimes the goal of anime isn't to produce a profit, it's to maintain viewership, or promote source (or branded) material, and sales of merchandise.