I played a couple of Wizards with an 8 on their spellcasting stat.
In 5e this affects the efficacy (chance of success) of almost all your offensive spells, because your spellcasting ability modifier applies to your spells DC and attack rolls. That means simply, don't focus such character on direct magical attacks, but build your tactics on buffing allies and manipulating the combat environment to their advantage.
Still, there are even a few direct offensive spells which are not affected by a low spellcasting ability score, for example Magic Missile and Sleep, and they are even scalable, so you still have options for direct attacks, just not the most optimized weapons as this kind of character is not supposed to have.
And of course an adventure is a lot more than combat, so focus spells which give you the spotlight in other pillars.
Now for the REAL downside I've encountered... that was the very low number of spells prepared! That is equal to your level + modifier, which was -1 for me, so I was stuck with only 1 prepared spell until level 3, yikes! That really sucked. I thought I would compensate with rituals (which a Wizard doesn't need to prepare) and that's when I learned there aren't actually that many rituals in the game besides stock stuff like Detect Magic or Identify (useful of course but not exactly exciting) so I had to be creative to get out as much as possible from Unseen Servant and Floating Disk. At some point I gave in a bit and just asked the DM to let me buy some scrolls...
I can see myself next time trying an absent-minded 8-Wis (or even less) Cleric. Domain spells being always prepared would help a lot, and there are for example many domains with channel divinity options that aren't based on your Wisdom score. For example I am pretty sure a War Cleric can be pretty powerful even with low Wisdom, although it would be a subpar healer.