Faced with mounting costs and declining enrollment, the Los Angeles Unified School District board approved a $7.5-billion budget Tuesday that will increase spending and lay off more than 100 library aides, clerks and other support staff next school year. The layoffs represent a growing acknowledgement by district officials that the nation’s second-largest school district has to downsize, adjusting to the reality of population loss caused by falling birth rates, gentrification and competition from charter schools. Supt. Michelle King’s budget plan, the second of her tenure, forecasts that L.A. Unified will continue to shrink. It assumes that enrollment, which stands at about 514,000 students, will decline by nearly 34,000 students, or about 2% a year, over the next three years. Such a drop probably would worsen the district’s financial problems because state funding is tied to enrollment and accounts for most of the district’s revenue.