Twist shifter cannot shift down – what might be the cause?

Without the shifter's resistance, the rear derailleur pulls towards the smallest cog (highest gear). This would also happen, e.g., when the shift cable snaps. (Rarer 'inverse' derailleurs would pull towards the largest cog.) To test the rear derailleur, pull the shift cable with your hand while turning the crank; if the derailleur moves and pushes the chain towards the biggest cog, it means the derailleur is ok.

If the (indexed) shifter doesn't click anymore, or if the internal catch doesn't seem to withstand the pull of the derailleur, it's most probably time to overhaul or replace the shifter.

It may simply be that old grease in the shifter has hardened up over time, so that the catch mechanism is stuck. This should be mendable if you open a cap of the shifter casing until a hole is revealed (used for threading in the cable), and spray in a generous amount of a light lubricant that will solve and flush out the sticky grease. WD40 is known to work quickly in such a job, but others also work well and may have better lubricating properties than WD40.

If lubricating doesn't help, it's possible that some internal part is worn out or broken and needs to be replaced. If spare parts are not readily available, it's a case for complete replacement.