Hi Tara!

One of the best ways to keep this patch healthy and lessen disease would be to prune it. The more air flow you get through the path, the better. Cut out old, diseased looking canes and this will not only help with air flow but can also stimulate growth! Beyond that, your best bet would be to spot spray with something like sevin which is a general insecticide. 

Blackberries are prone to many diseases if they are not managed regularly. I am doubtful that the webs you see on your plants are a pest--I think they are spider webs. Those spiders might also be beneficial--eating small insects that feed on your plants or fruit. We never recommend using pesticides without a proper ID first. It is unlikely that what is creating the webs is killing branches of your blackberries.

Blackberry plants frequently loose canes from cold injury during the winter. Older canes are also less productive and may simply senesce. 

Here is a link to our website on fruit. 

Here's an article on raspberry cane borer (I have seen its damage on the wild raspberries and blackberries). 

You can mow the blackberries, but if you do it now, you will not get any fruit this year and probably not next year. I would also recommend removing the debris after they have been mowed (raking and bagging or burning), because fungal pathogens can remain in the cut stems and re-infect the new growth next season. 

You can read about pruning blackberries on this website. 

Here's a great article from Minnesota about brambles. 

I would recommend hand pruning at least a few of the plants. Maybe select the easiest to reach and take care of them. You could mow the rest or let them go.