For a behind-the-scenes answer, it's because the filmmakers at the time had not thought of putting them in the film because, at the time of filming in 1989, mobile phone technology was still in the early stages of development and widespread adoption. At one point we see someone using a public phone to call with Doug, who is only a few yards away, so this would imply that mobile phones indeed do not exist in this future world (unless that person used a public phone out of fear of being traced or tapped).
In the context of the film, mobile phones may have fallen out of use as people began living on other planets. At its closest, Mars is four light-minutes from Earth. This distance would create such latency in the call that any real-time conversation would be next to useless. Instead it seems that people have switched to less mobile but longer range devices, such as that seen in the car which allows Richter and Cohaagen to communicate almost instantly from Earth to Mars. However, this wouldn't explain why so few characters use any kind of portable device to communicate with others locally (i.e., on the same planet). For whatever reason, such a manner of telecommunication is not a staple of the customs of the depicted futuristic societies. The same thing can be noticed in
Back to the Future Part II (1989) when Marty McFly Jr. uses a payphone in 2015.
ANSWER: That's like asking about a Jules Verne novel about space travel (From the Earth to the Moon 1865), why there aren't any microwave ovens? A futuristic movie is based on whatever future is perceived from that point when it was written, and to fit the story, having nothing to do with the actual future and what THAT future holds.
ANSWER: Back To The Future part 2 which was also filmed in 1989, had a payphone (which MartyMcFly Jr. uses). Same explanation: Mobile phones were not anticipated (Radiotelephony 1918 Goldsmith) to be so widespread and the extinction of payphones was not forseen. That film also had hoverboards, which as of 2021, do not exist.