This is a discussion on light stay red within the Converter Boxes and ATSC Tuners forums, part of the Over-the-Air (Antenna TV) category.
What brand of box? assuming it's Best Buy it should be an Apex DT-250. the power light is red when off(or in Analog Pass-through mode) and green when on. first thing is try using the front panel button. if that doesn't work, try unplugging the box, waiting about 10 minutes, then plug it back in. for some people (especially those in areas with horrible power grids) the boxes get 'confused' or even 'lock up' just like a computer. seeing as they do have a type of computer CPU (an ARM CPU) and a simple Operating System stored inside a ROM, they are subject to that. the unplugging trick basically does a 'hard reboot' of the box's firmware and it should restore it.
If it's just the remote, the batteries which come with the boxes are horrible and some i have had come with packaged batteries already leaking acid. i'd toss them out and go with higher-quality name-brand batteries.
If it is in Analog Pass-through mode that basically turns the box off. try either using the front panel button or using the button for Analog Pass-through on the remote, which should turn it back on in digital mode.
The Apex DT-250 has issues with bad power supply windings which can short out but if that happened you'd get no power at all. not even the red standby LED inside the power button.