LAX: The closest international airport to Pasadena is LAX (Los Angeles International Airport). This is likely to be the best option for most international travelers, and may be preferable for domestic travelers as well because it offers the widest range of flights.
ONT: A second international airport is ONT (Ontario International Airport). This airport is smaller and easier to get in and out of, but it is farther away. However, if you are planning on renting a car, then ONT may be an option for you, and flights and car rentals may be less expensive than via LAX.
SNA: A third international airport is SNA (John Wayne Airport). SNA is further away than LAX, and during peak travel times, the main highway (I5) between SNA and Pasadena is extremely clogged with traffic. It is best to avoid the roads completely in the time ranges 07:00-09:00 and again 15:00-20:00. Also, the highways immediately around SNA are somewhat convoluted; it's a good idea to plan your route ahead of time.
BUR: For domestic flights, a fourth airport to consider is BUR (Bob Hope Airport). It is by far the closest airport to Pasadena, but serves only a limited number of airlines. (If you fly Southwest Airlines, however, it is probably your best bet.)
For direct and economical transportation to and from the airport, an app-based service is your best bet. Estimates for one-way rides using Lyft are:
Shuttles are also available. SuperShuttle rates between LAX and Pasadena are roughly $30 for a standard, multi-stop shuttle, and $60 for a direct, shared ride. https://www.supershuttle.com/locations/