From the pictures it is not possible for me to determine with certainty what the problem is. One possibility may be sun burn. Hostas can be sensitive to the sun which is evident if the leaves look faded, burned or are deteriorating from the edges inward. There is certainly some evidence of that in the pictures but I have is no way of telling if the location receives intense sunlight or if other plants clearly not in direct sunlight have the same characteristics.
Another possibility is draught stress. This results in the leaves turning brown along the edges. Keeping the soil evenly moist but not flooded is the best preventive. Watering with an inch of water weekly and covering the soil with mulch is recommended.
Browning of the Hosta leaves may also come as a result of plant disease called anthracnose. This disease produces some tan or white spots on the Hosta leaves with the borders turning brown in color. Fortunately, anthracnose does not infect the plant permanently; removing infected foliage and organic ground clutter in the fall prevents the reappearance of symptoms in early spring growth.
For a definite identification I suggest taking a sample to the The Sussex County Extension Office at 16483 County Seat Highway, Route 9, west of Georgetown, Delaware. The sample should include healthy leaves and leaves with both healthy and affected sections.
Sorry Joan (not Joah) my typing needs help.