The White House released a $4.7 trillion budget Monday that proposed sharply reducing spending on safety-net and many discretionary government programs, while boosting defense and border-protection funding, in a plan that also projected a long run of continued deficits.

While the new budget isn’t likely to become law—Congress ultimately writes spending bills that the White House can approve—it lays out an executive branch vision for government. This one included shifting priorities for government spending.

“This...