The answer is no. Indeed, let $X\sim U(0,1)$, where $U(0,1)$ is the uniform distribution on $[0,1]$, and let $Z$ be an independent copy of $X$. Next, let 
$$Y:=\frac Z2\,1(X<1/2)+X\,1(X>1/2).$$ 
So, the distribution of the random point $(X,Y)$ is the half-and-half mixture of the uniform distribution on the square $(0,1/2)^2$ and the uniform distribution on the diagonal $\{(x,x)\colon1/2<x<1\}$ of the square $(1/2,1)^2$. 

Then $Y\sim U(0,1)$. So, the common density of $X$ and $Y$ is nonincreasing on $[0,1]$. 
 
One the other hand, $M=\min(X,Y)$ has the non-monotonic density $g$ on $[0,1]$ such that 
$g(m)=2-4m$ for $m\in(0,1/2)$ and $g(m)=1$ for $m\in(1/2,1)$.