The answer to this question changed by the month and could be very different from place to place. On January 24, 1848, James W. Marshall discovered gold in Sutter Creek, California, making the year - and the one that followed - pivotal for the new territory. The full effects of the discovery, however, was slow to be felt in 1848. As a result, it is not entirely appropriate to talk about "boomtowns" in that year - or at least in the beginning of the year.

The California territory was changing in the wake of the U.S. assuming control of that part of the Pacific Coast following the Mexican-American War (1846-1848). At the same time, that change was slow to affect day-to-day life. Much of California remained the same, and settlements tended to have deep roots. The governmental structure changed - at least officially - with the end of the war in February 1848, but the one thing that is clear about territorial government in the West is that there was a tendency to work very hard and very quickly to achieve organization and stability. There were three layers of law enforcement that was or came to be available: federal marshals, county sheriffs, and town police. Once a county or town government was organized, it would very quickly appoint law enforcement. If a sheriff were to die, the county commissioners would appoint a replacement. If the town were of sufficient size and stability to warrant incorporation with governmental oversight, it would quickly have a police department (unless there was an agreement between county and town government). At the point of a chief of police dying, his replacement would be appointed by the town fathers, again, very quickly.

The Gold Rush of 1849 changed things with the influx of tens of thousands of immigrants. In this case, however, we must remember that the gold deposits were extensive - scattered over a large swath of land, and so the new arrivals tended to be scattered without towns. In this case, the county sheriff had authority, but because of the incoming population, it was difficult in the first year or two for government to keep pace with the growth. The county government would replace your dead sheriff, but the sheriff's department might lack the manpower to achieve proper policing of the large numbers of new arrivals, so there was a degree of chaos for the first year or more.

I suspect that what you're really asking about with the term "boomtown" is the phenomenon that occurred in the intermountain mining West, particularly after 1859. In the hinterland of the mining West, precious metals tended to be intensively located, meaning that discoveries located seams of ore that went underground. This caused a dramatic demographic shift for the American West from placer, extensive mining to underground (or quartz/hard rock) intensive mining. For the latter, miners were brought together in urban settings - your "boomtowns" - and they tended to be employees of corporate mining, unlike the classic '49ers who worked for themselves in small groups. In the case of these boomtowns. The communities sprung up suddenly and tended to grow and were often abandoned quickly if the ore proved limited in quality or quantity. In this case, again, it could be difficult for territorial government to keep up. With the 1859 discovery of the Comstock Lode - one of the richest silver and gold ore bodies ever discovered - Utah Territorial government had difficulty extending order to the far west of the Great Basin, far from the capital in Salt Lake City. Despite this, Virginia City was incorporated as a city very quickly under Salt Lake's authority, and the city was able to employ a police department to prove order for the new growing community, particularly needed since territorial and county governments were far away. When the Nevada Territory was organized by mid 1861, the territorial legislature quickly reorganized the western Great Basin with more counties so that their governments were closer could address the very question you ask - how to provide law enforcement on the county level even when your sheriff had died.