How do you insulate a basement window that has a window well?

You can use a window insulation kit from Three M or bubble wrap

Does the basement window have a wooden frame? My 100-year-old house had the same issue. Caulk was better, but a second window made of glass in a wooden frame was best. I routed out some pine board, set in the glass in the frame and then attached the frame to the inside of the window.

Look for Magnetic window installer have him measured the window to have proper size to cover window from inside the best thing it is removable for cleaning it is very efficient.

There has been a post with an idea on this problem - Get bubble wrap. Mist the window a little with water - DO NOT SOAK IT!! - then take bubble wrap and put the flat side against the glass, bubbles in the house. Smooth it down and wrap will cling to window, adding insulation and allowing light to enter.

If the window is not operational, and not needed for emergency access - Have a piece of plexy glass cut to fit. An old fashioned hardware store or glass store may be able to do this for you. Place that on the inside over the old window and caulk it in place. Or if looks are not important - just duck tape it in place over the old window. You will still get light.

Can you just afford to replace the basement windows only? Just choose a generic style that will not matter if it matches exactly to the rest of the house.

Make a timber frame and fix Plexiglass or Plastic sheeting to it. Then fix that to frame eith Screws, Double sidded Tape or Velcro.