Of course $\{ a^nb^n \mid n\ge 1\}$ is a DCFL. Adding $a^*$ to a DPDA for that language is straigthforward: as long as we did only read $a$'s that state we are in is final, after we read a $b$ the states are as in the original DPDA.
In this way we can show that the family DCFL is closed under union with regular languages. The construction simulates the original DPDA with a (deterministic) finite state automaton in parallel, using a product construction.
But there is a caveat. The original DPDA is deterministic, but it might end some of its computatuins in an infinite $\varepsilon$-loop. This will infinitely interrupt the simulation of the parallel finite automaton. Hence we need the requirement that the DPDA we start with has no such $\varepsilon$-loops. That is non-trivial, but as a normal-form it has been obtained in the construction for closure under complement for DCFL.