Since they opened the Gotthard tunnel through the Alps. Is it faster to go from Munich to Genoa by train or by car?
I doubt it will change much, driving will still be faster. Based on a Google Maps itinerary search, we find out that it currently takes up to 7 hours by car, while it takes 9 hours by train, through Austria.
The Gotthard tunnel is located in Switzerland, and while its goal is to reduce the time to travel between Zurich (Switzerland) and Milano (Italy) to 2h40, there is no fast train route between Munich and Zurich.
What is interesting to notice is that on the map, by drawing a straight line between Munich and Genoa, it does not go through Switzerland (or only along its eastern border). So going by train through the Gotthard tunnel will likely not help.
If we still try to estimate the time it takes between the two cities with the estimate given in the first article, going between Munich and Zurich currently takes at least 3h40 according to the Deutsche Bahn in a bus (!), then say 2h40 to Milano, and finally with the Google Maps search we found that the train between Milano and Genoa takes 1h45. By adding up all of them, we still find 8 hours.
In the end, even if somehow the Deutsche Bahn or the SBB decided to run a train all the way from Munich to Milano/Genoa, the total time it takes will probably not beat the driving time. So unfortunately there will be no change in the hierarchy of faster transportation, driving will still be faster.
Google Maps suggest 5,5 hours driving time from Munich to Genoa ( a bit optimistic since it's estimated "without traffic" which is like, never). Travel time by train (as per Deutsche Bahn Website) is 11-12 hours.
The new tunnel is said to reduce driving time from Zürich to Milan by one hour. I don't think it will have much more effect on the driving time from Munich to Genoa.
The main purpose of the Basistunnel is to remove freight traffic from the streets (more friendly to the environment plus by the now the Swiss are rather annoyed by constant truck noise). Reducing travel time was not a big consideration.
Openstreetmap’s car route Munich–Genoa does not tangent the Gotthard area, and neither does bahn.de’s train route suggestion for that relation. The latter does Munich–Verona–Milano–Genova with a EuroCity connection across the Brenner to Verona and takes ten hours.
Considering that to use the Gotthard base tunnel by rail you would need to somehow get to its northern portal, the options of coming close are either via Bregenz or the grand detour via Stuttgart/Karlsruhe. (The latter is funnily suggested as an overnight alternative using a late-night ICE from Munich to Karlsruhe. The former is only sort-of found in form of a connection to Friedrichshafen and then a ferry across lake Constance. It is also a late-night/early morning connection.) The grand detour is unacceptable by day because of how slow the Ulm–Stuttgart strech is (and still will be until the new track has been built). The Bregenz variant is also not truly one because of the rather slow transit through the Allgäu region down to the lake. I don’t think the EC to Zurich will provide a faster option from 2017 onwards – it takes 4:20 just to get there.