No. If they did, then they would still generate the mapping class group of the closed surface that results from gluing a disc to the boundary component. However, in that surface they all commute with the hyperelliptic involution, which is not central for $g$ at least $3$.
In fact, Humphries proved that you need at least $2g+1$ Dehn twists to generate the mapping class group. This is contained somewhere in Farb-Margalit's primer on mapping class groups.