train travel Mannheim - Basel ~~ do we need to book ahead?

You could show up last minute and buy a flex price ticket since seat reservations are not obligatory on German trains. However, the fare will be approx. the double of a pre booked saver fare. Saver fares are train-specific, flex price tickets are not. But as soon as you want a seat reservation you have converted your "flexible" ticket into a train specific one ... ;)

But since a 2nd class seat reservation is only 4.50 EUR per person, it won't break the bank if you don't make it. The seat reservation expires 15 minutes after the train departs, so if the train is crowded, others will be able to use your seats if you are a no-show.

So even if you bought a ticket, you do not have a guaranteed seat in 2nd class unless you buy an additional seat reservation. 1st class tickets include seat reservations.

Prices and tickets at www.bahn.com

As sla says, its not a matter of reserved seats, its a matter of price. Buying today will be a lot cheaper than buying on the day at the station.

Current price on https://www.bahn.com/en/view/index.shtml for Monday 9th September, buy today at €19.90
Price for today: €63, apart from one train which is fully booked.
And seat reservation is in both cases an optional extra, but recommended.

And seat reservation is in both cases an optional extra

Not exactly, it is included in a 1th class fare if you book the seat together with the ticket. If you forget about that, it's €5.90 in addition - seems the work done by the DB computer is €1.40 more worth for 1th class (but, in my experience, complaining in a Reisezentrum helps - they will add the forgotten reservation free of charge) ;)

Chris F just identified a quirk in the DB ticketing system. On Sept 9, a ticket from Mannheim Hbf to Basel Badischer Bf on the 10:36 ICE is 19.90 EUR for a 2nd class Super Sparpreis ticket. But if you specify Basel SBB station as the destination, the price is 39.90 EUR. So they are charging you 20 EUR to go across town in Basel. Even at Swiss prices, that is a bit excessive, or a spectacular deal, depending on how you look at it.

Sam, thanks, I hadn't spotted that. I just put in Basel Badischer Bahnhof because that is the DB station. The leg from Badischer Bahnhof to Basel SBB is officially cross-border, even though it is within Basel, so different rules apply and you sometimes get anomolies.

And the fare between the two stations is CHF 3.80. That fare is valid from anywhere in Basel to anywhere else by any train, tram or bus.
AFAIK that is the cheapest fare for an ICE anywhere ☺

Chris F just identified a quirk in the DB ticketing system. On Sept 9, a ticket from Mannheim Hbf to Basel Badischer Bf on the 10:36 ICE is 19.90 EUR for a 2nd class Super Sparpreis ticket. But if you specify Basel SBB station as the destination, the price is 39.90 EUR.

@Sam
Curiosities such as these are not a quirk of the DB system but a systematic feature of the European Railway Tariff (TCV) to the disadvantage of travellers whose destinations is just beyond a border.

Just another example: Munich Lindau EC saver fare tomorrow = €19.90, Munich Bregenz on the same train = 29.90. Buying a ticket to Bregenz in the Lindau railway station: from the DB machine = €5.20, from the ÖBB machine immediatly next to it, but "beyond" the imaginary border = €3. Until recently, it even made a difference whether you bought a ticket from Munich to Bayrisch Eisenstein (DB) or to Zelenzna Ruda (CD), even though both stations share the same building. That latter has been fixed in the meantime; actually a pity, somehow. ;)

Thank you to all of you. this information has been very helpful to me, who has never used trains in Europe before.