Before you fly
When gate agents are deciding whom to upgrade, often it's the passengers who have elite status or have paid full fare who get priority. If you fly a particular airline frequently, contact the airline ahead of time and ask what's required to get this elite status. You never know; you might qualify for an upgrade, which means a little more room. And that may mean less contact with germs -- which leads to a decreased chance of catching an illness on the flight.
At the gate
Politely approach the desk and ask how full the flight will be. Also ask if there are free or discounted seat upgrades available. Remember, not only are first- and business-class seats more comfortable; they're further apart, giving you a safer amount of airspace between you and your fellow passengers.
If you're not willing to pay for an upgrade, ask to be considered for a seat left behind by those who do upgrade. Often, the seats these passengers are leaving behind are preferable to the one you have, particularly if you booked at the last minute.
Many people are concerned about sitting next to toddlers and small children, particularly if they have runny noses or are drooling. If you notice passengers with young children in the boarding area, mention to the person at the desk (out of the parents' hearing) that you'd prefer not to sit next to a child and that you'll be requesting a move if you're seated near one. You aren't required to explain your reasons.
On the plane
If you notice someone coughing or sneezing near you, immediately begin the process of requesting a seat change, as this can take some time. You can choose to be honest with the flight attendant, or you can offer another reason, such as preferring to be in another part of the plane.
When scouting for free seats, look to see what's happening in the exit rows over the wings. Families with young children are moved from these seats due to emergency requirements, and sometimes a reorganization means that the seats become available.
If someone appears to be truly ill, you're within your rights to bring the situation to the flight crew's attention. Airlines are becoming stricter about allowing sick passengers to fly. One case, in November 2009, made headlines because a woman was pulled off a United Airlines flight to Hawaii because she was visibly sick to her stomach.