Certified GKH Gardening Expert
Go ahead and prune it back. Get rid of all the old dead canes. Perhaps do a shaping pruning if it has gone a bit all over the place on you. Climbers sometimes do that. Once pruned back, seal the cut ends of the canes with some White Elmers Glue or the Tacky Glue from Craft Stores. The glue cap on the canes will help keep cane boring insects and wasps from causing harm. When you water the climber mix some Super Thrive in with the water, water with the Super Thrive in the water the next 5 to 6 times the rose needs watering.
For most roses, yes, you do need to prune them back every year. Old Climbers bloom only on the "old wood" (the previous years growth), so wait for them to start leafing out before doing any pruning on them. That way the rosebush shows you where to prune. Pruning the old climbers too soon will eliminate many blooms. https://www.gardeningknowhow.com/ornamental/flowers/roses/pruning-climbing-roses.htm