If you will be flying to us, Oakland International Airport (OAK) is the closest airport to Oakland and San Francisco International Airport (SFO) is another option. You can take Uber/Lyft or public transportation (BART) to Berkeley or rent a car.
If you choose to rent a car or drive from where you live, you'll be able to get around the Bay Area well. Parking in the East Bay is especially doable. That said, please be aware that car break-ins are very common in the Bay Area and you should never leave anything valuable in your car, or anything that would attract theft (especially no backpacks/bags, luggage, etc.).
In addition to some nearby hotels, there are also many AirBnB options in Berkeley and Oakland.