The trouble with Jasmine officinale is exactly what you've described if you can't give it enough room - left to its own devices, with plenty of scramble room, it will get 35 feet high and about 12 feet wide, and grown like that, its at its best. The dead looking twiggy parts are covered by new growth a little later in the season, but unfortunately, you need to keep it a certain size and under control.

The time to thin the shrub is actually after its flowered, when you cut back half the stems and take out any dead or diseased parts, and tie in remaining growths. If you do a hard prune now, you may not get much flower later, but you can prune out dead parts if you like. Whether you tidy it now is largely your choice...

Oh dear Bamboo, my husband attacked it today whilst I was out at work. He thought the dried out stems were all dead and cut off all the flimsy woody bits. Thanks, I'll remember next time to cut back after flowering.....

Well, the pruning certainly won't kill the plant, and if he just cut off dead bits, it'll be fine. Those dead bits usually remain, but simply get covered over by new growth from elsewhere.

He gave it a good pruning too, but maybe it will still flower.....