The best time to travel depends on your needs and health.
The river locks in Europe tend to do their maintenance around April, so you run the risk of catching a bus for some of the journey and moving to a new ship.
Summer, autumn/fall cruises
Summer in Europe is a mixed blessing. It is warm and sunny from late June through September, so this is a great time to take a river cruise. Many attractions are open only in the warmer months and the daylight time helps you to sight-see in natural light beyond 9 pm.
The rivers and canals are not very deep, so the river ships tend to have a shallow draft, less than 2 metres. Lots of sunny days means little or no rain, so the river level drops and after a certain point the ships cannot travel without hitting the bottom. At the other extreme, lots of rain means the river levels are so high that the ships would hit the bridges if they tried to move.
The usual outcome in these extreme cases is that the ship will dock and passengers will have to sight-see by bus. Every year at least one cruise will experience one of these extreme situations, so be prepared.
Christmas Markets are the main attraction for those wishing to travel in late November and December. The best time is 24 November through 23 December, when those markets close. Some Christmas shops are open year-round and some open in early November.
The “Christmas Markets” cruises are the same as the standard ones, but you simply have the benefit of seeing the Christmas stalls during your free time. For example, you can choose not to go back to the ship for lunch but instead spend that time wandering through the markets.