Sensing a tag passes through a virtual "fence"
Unfortunately your requirements collide with your price expectations. Most active tags are heavy enough to annoy the cats. Passive tags have either too short range (e.g. 10 cm for RFIDs) or require high radiation readers (you don't want those in your house 24/7). And the readers are relatively expensive.

So, save money and go with IR photointerrupters as was suggested above. I'd put them in groups of 3, two near the floor to get direction and one higher to detect humans. Cheap and easy + some interesting programming.

Alternative would be to use PIR motion detectors. These give you a bit more flexibility in placement around house than photointerrupters. Ready made PIR modules can be found for just a couple bucks.