Rhine River Bern to Amsterdam

I would not expect to find any single ferry covering that whole distance because there's no market for it among locals. Most people are in a hurry and would take a train or fly. I don't know the features of the various, tourist-oriented cruise companies.

Train can be done with as few as two connections (such as Basel and Frankfurt) or you could stop and take the K-D Line ferry somewhere between Mainz and Cologne.

You should probably incorporate into your trip a half-day cruise through the truly scenic part of the Rhine. Bingen (just west of Mainz) to St. Goar or Boppard (just south of Koblenz) would be good.

Bingen: https://media.holidaycheck.com/data/urlaubsbilder/images/41/1156171934.jpg
Braubach: http://www.wald-laeufer.de/fotos.bilder/20090614/img_20090614_011/Marksburg-und-Braubach-am-Rhein.jpg

In between are similar old-world towns worth stopping over in - Bacharach, Oberwesel, St. Goar, and Boppard have cruise boat stops. You could stop over at one or more of those places.

Here's an idea with 2 nights in the scenic part...

• Take the train (or drive) from Bern to Boppard, drop bags at hotel, see town, take the chairlift ride, have dinner.
• Next morning: Take train (9:07) south to Bacharach (9:29), see town for an hour. Board 10:30 southbound train for Bingen (10:43), board 11:30 boat north to St. Goar (12:55.) See Rheinfels Castle. Cross river by ferry, board northbound train (14:32) for Braubach (14:53), see town and Marksburg Castle (16:00 English tour in summer months.) Return by train to Boppard in evening.
• Next morning: visit Oberwesel or other town(s) by train or car. Move on to next destination.

KD Cruise boat schedule (probably similar for 2017.)

I see Laura has just posted as well... 2 heads-ups to you:

1.) The "KD RHINE" website linked to the schedule she provided sells a day pass for cruising - but it's normally "overkill" in the sense that you can cover the best scenery (like Bingen - St. Goar, which I suggested) for half the price using regular tickets bought at the dock. The day pass is the sort of thing you might want only if you were doing a much longer trip that also included less scenic sections (like Mainz-Bingen and Boppard-Koblenz or Koblenz - north.) In this situation the day pass may be a better deal, but check the regular prices first.

Here's a map of the Bingen-St. Goar segment, btw: http://www.bingen-ruedesheimer.com/images/map/rhine-valley290.jpg

2.) Also, I referred to the "KD cruise boats" whereas Laura has used "ferries" for those same vessels - not a problem except there are also different ferries that have nothing to do with the cruise boats - like the St. Goar ferry I suggested, which just shuttles across the river all day long.

http://www.schiffbilder.de/1024/personen-fahrzeugfaehre-loreley-vi-zwischen-18381.jpg

(There are also ferries like this in Bingen and in Boppard.)

Hansen, you can't start your cruise at Bern. The cruise boats (the ones you sleep on) run between Basel and the sea at Amsterdam or Rotterdam. Bern is on the Aare, which is a tributary of the Rhine, but the Rhine is only navigable as far as Basel, Basel has Switzerland's only docks. Between Bern and Basel only certain sections are navigable for small boats only. Bern to Basel takes about 1 hour by train.

The big cruise boats run from Basel to Amsterdam, but you don't have to do the complete stretch, you can do part of the route. These cruise companies provide guided tours of the various stops - I know because I keep on coming across the groups having guided tours when I walk around Basel at the weekend. Judging by what I see, they provide coaches (buses) to take you between the boat and the sights, no long walks involved.
You will always be going around in groups, and always with an English speaking guide, which you may consider a pro or a con.

The most famous and most spectacular section is the Rhine Gorge between Bingen and Koblenz. As others say, on this section there are also day boats, run by the KD-line, which you can use to hop from town to town up and down the river. There are also two rail lines which run along the river, one on either side, which give good views. There are no bridges between Mainz (above Bingen) and Koblenz, the only way from one side to the other is the KD boats or ferries direct from one side to the other.

Depending on the time available, your budget and your preferences, make your own choice.

Just to elaborate a little on what the others have hinted...

The Rhine between Basel and to around Wiesbaden could be just about any river anywhere else in the world. Not that the scenery is necessarily boring, it just isn't particularly unique. The stretch around Mannheim and Ludwigshafen looks somewhat like a dystopian vision of the future. Around Wiesbaden, things start to get a little more interesting, then between Bingen and Bonn (with the exception of a rather ugly industrial strip immediately north of Koblenz), the river goes through the famous gorge with all the scenic towns, castles and such. Most of the river downstream of Bonn is industrial.

Our daughter-in-law is a German national from a small town just west of Speyer in the Palatinate. We have taken family outings on the Rhine from Mainz several times. This takes you through a good portion of the castle area and by Die Lorelei. You can take it back or catch a return train at one of the stops (Bacharach, St. Goar, etc.). Makes a great day trip.

"We have taken family outings on the Rhine from Mainz several times. This takes you through a good portion of the castle area and by Die Lorelei."

Yes, it is possible to start in Mainz. However, there are just 1 or 2 boats per day from there, and starting there means nearly 2 hours of cruising through flat, mostly unremarkable territory before you reach the scenic castle area (Ruedesheim/Bingen and north.) The extra cruising of course costs extra. Also, the boat dock is a hefty hike from Mainz train station. Alternatively, most people choose to take the train to Bingen (only 20 minutes from Mainz via RE train) or R'heim and board the cruise boat there. There are typically 5 boats per day from these towns, and the stations are closer to the boat docks.