Blade Runner was seemingly produced by 'Filmways'. This production company was founded by Martin Ransohoff in 1960. Since then it has grown and purchased several other smaller companies, such as 'American International Pictures'.
Along with Bud Yorkin, Jerry Perenchio owns the rights to Blade Runner, as his bond completion company took ownership of the film when it went over its budget. The production company helped it become one of the first ever DVDs. Unfortunately, due to bad quality, the copies stopped selling. For many years, Perenchio refused to allow any new DVD edition of Blade Runner, including a planned 2001 Special Edition, to be issued.
There is very little information on the actual production companies that made it. I struggled to find much. Sorry, I'm not entirely positive the Filmways was the proper production company, but it is displayed as such on several websites. As for the actual company, there is very little.