BLM has about 200 law enforcement officers to patrol 258 million federal acres. Needless to say, they are totally understaffed. There primary mission may have once been resource protection in the "classic" sense, which would include enforcement of off road vehicle rules. Unfortunetly, the number of serious crimes occuring on Federal lands has increased dramatically. Clandestine drug manufacturing facilities, sophisticated marijuana grows, dumping of toxic and other hazardous wastes, etc. These types of investigations require more and more of their time and resources and will often involve outside agencies. This is just one example of how one agencies resources are stretched thin.
To answer your specific question, enforcement of vehicle violations will usually occur by the agency that administers a specific area, usually these officers would be considered "Rangers". Some OHV areas are managed by BLM (Federal), State, or even Local governments, so it may vary greatly depending on where you are.
This is not to say that the local law enforcement agency cannot be directly involved in the enforcement actions on these lands. For example, the Imperial County Sheriff and the San Diego County Sheriff have a large off road enforcement detail they muster for holiday weekends to control up to 50,000 people who gather in Glamis, which is managed by the BLM. This multi-agency approach is common, since BLM couldn't possibly patrol this many people at one time.
For the most part, state laws are applicable to these areas and nothing precludes a local agency officer from enforcement in these areas as well. In order to be effective, local agencies must work closely with state and federal agencies in order to provide effective coverage.
Some agencies will have jurisdiction over specific types of crimes no matter who the managing agency is.
As Paul mentioned, Search and Rescue is usually a function of the County Sheriff, but this does not preclude other agencies from having their own groups. All but the smallest Counties will muster a SAR group. It wouldn't be uncommon for there to be a SAR coordinator for the State either, as Paul said, to coordinate larger responses.