Probably the most effective and longest lasting will be Bayer Ground Clear, but you must be sure that it cannot leach sideways onto soil where you are growing plants. It's systemic, so it works 'through the green', is most effective on smaller weeds, and best applied between April and September when the weeds are growing strongly and there's enough green growth. Can be either watered on in a drench or sprayed, and should be effective for a year, particularly if used towards end of April. Larger, perennial weeds will probably need more treatment though.
The only permanent option I can think of is to cover the area with thick black plastic and peg it down. That should stop everything, but it may not be practical to carry out.

Have you considered planting low growing spreading evergreen perennials or dwarfish evergreen shrubs(if there's enough soil to grow them)

Once they grow together there will be virtually zero maintenance as weeds will be supressed.

Winter flowering heathers are very effective for weed supressing

If he absolutely must climb the fence why not get him to tie a rope round his waist and anchor it to something to stop you worrying? He may not agree but its worth a try.

Hi, Steragram's suggestion should be made compulsory, you certainly wouldn't get me standing on a 4ft wide strip of land, with a 60ft drop, without a safety line attatched very firmly, Derek.

Thanks to everyone for their helpful comments. I think I'll go for the Bayer Ground Clear - sounds favourite to me. IT is good to get all the suggestions.