As a general rule, when travelling within the Schengen Area, you will not have to pass through passport checks. Member countries are allowed to set up temporary border checks, but these are usually placed on road and rail border crossings, not in airports.
Virtually all airports in the Schengen Area (especially the ones with scheduled flights!) have separate Schengen / non-Schengen sections. Passengers departing to / arriving from outside the Schengen area are routed through passport checkpoints, passengers flying within the Schengen Area are not. A passenger flying from Vienna > Frankfurt > New York will pass the passport checks in Frankfurt (not in Vienna), before boarding the New York-bound flight. Sending passengers flying within the Schengen Area through passports checks would be a big violation of the Schengen Treaty. It does not routinely happen, possibly only if the Schengen section of an airport is out of service for some unforseen reason.
Now, there's one important point to consider: not all airports are created equal! I flew many times between Vienna and Zurich, and there were absolutely no checks on either side. No one asked for my ID at the security controls. There are automated check-ins and boarding gates on both airports. All I needed was a boarding pass with a QR code (printed or in a mobile app). It would be relatively easy for someone else to fly on my ticket (I'm not encouraging that!).
However, when I flew with Ryanair from the Bratislava airport to Berlin (both within the Schengen Area), it was a completely different story. I was asked for my boarding pass and my ID a total of four times: at the check-in, at the security, at the gate and inside the airplane. In three cases those were airline employees asking the documents, so there is a reasonable chance of them accepting a driving license or whatever. But the security checks in Bratislava are performed by regular Slovak national police officers and they for sure won't accept a driving license (no even if it's issued in the EU) as a proof of identity. That doesn't work in Europe, except for the UK, which is not a Schengen country.
What is the situation like in Basel and Porto? I don't know. More and more airports, especially the bigger ones, have automated check-in and boarding, and outsource the security checks to private contractors. Additionally, in some cases such as Zurich (which could be similar to Basel), the security is carried out by police officers, but they don't ask for IDs of any kind (unless of course you're caught carrying something very dangerous).
In all likelihood, you may successfully fly to Porto and back without your passport. However, I would not advise that. Some countries (Italy, for example) have strict laws on providing accomodation and you might not be admitted to a hotel without an official identity document. Such problems are usually solvable, but it's a pain, not a pleasant holiday. More importantly, travelling in the Schengen Are without an official ID is illegal (you probably already know this). If the police asks for it and you can't produce it, you're very likely to face a fine and/or even detention. Again, not worth it. In an extreme case, since you're not an EU / Schengen country citizen, your Schengen visa might be revoked and/or you can have trouble getting another one after the current one expires.