Generally, all states prohibit driving while under the influence of drugs or alcohol. But state laws differ in how a DUI offense is defined. Some states define DUI in a way that includes intoxicated biking. Here are some of the basics on how a person might be arrested for a DUI while on a bike.
In many states where you can get a DUI on a bike, it's because of the way the state DUI laws define "vehicle." In many states, DUI laws apply only to motor vehicles, which doesn't include motorless bicycles. But in other states, DUI laws cover all vehicles whether motor-driven or not. Under these more expansive laws, it's possible for a person riding a bicycle to get a DUI.
For example, Colorado's DUI laws prohibit driving while under the influence of a "motor vehicle or vehicle" and the definition of "vehicle" includes bicycles. Similarly, in Florida, courts have interpreted the DUI law—which refers only to driving a "vehicle" while under the influence—to include bicycles.
A few states, such as California, have laws that specifically prohibit biking under the influence. In these states, violators would presumably be prosecuted under the drunk biking law rather than the DUI laws that apply to motor vehicles.
Generally, the defenses to a bike DUI charge are the same as those that could apply to a standard motor vehicle DUI. For example, a cyclist who's charged with a "per se" DUI for biking with a blood alcohol concentration (BAC) of .08% or more might have a defense based on the reliability of a breath or blood test result. (Depending on the laws of the state, a cyclist might have a related argument called the "rising-blood-alcohol defense.") In other cases, a cyclist might have a viable defense based on an unlawful police stop or illegal search and seizure.
If you're charged with biking under the influence, you might find prosecutors to be more willing to offer a favorable plea bargain than they would be for most other DUI offenses. Protecting public safety is the primary purpose of DUI laws and bike DUIs would normally pose less of a risk to the public than motor-vehicle DUIs. So, presumably, prosecutors might be more lenient with DUIs where the offender was on a bicycle.