We study degree-penalized contact processes on Galton-Watson trees (GW) and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex $v$ with degree $d_v$ infects its neighboring vertex $u$ with degree $d_u$ with rate $λ/f(d_u, d_v)$ for some positive function $f$. In the case $f(d_u, d_v)=\max(d_u, d_v)^μ$ for some $μ>0$, the infection is slowed down to and from high degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts. We show that new phase transitions occur in terms of the parameter $μ$ (at $1/2$) and the degree distribution $D$ of the GW tree. - When $μ\ge 1$, the process goes extinct for all distributions $D$ for all sufficiently small $λ>0$; - When $μ\in(1/2, 1)$, and the tail of $D$ weakly follows a power law with tail-exponent less than $1-μ$, the process survives globally but not locally for all $λ$ small enough; - When $μ\in(1/2, 1)$, and $\mathbb{E}[D^{1-μ}]<\infty$, the process goes extinct almost surely, for all $λ$ small enough; - When $μ<1/2$, and $D$ is heavier then stretched exponential with stretch-exponent $1-2μ$, the process survives (locally) with positive probability for all $λ>0$. We also study the product case $f(x,y)=(xy)^μ$. In that case, the situation for $μ< 1/2$ is the same as the one described above, but $μ\ge 1/2$ always leads to a subcritical contact process for small enough $λ>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.