A property $Π$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $Π$, every superset $Y \subseteq U$ of $X$ also satisfies $Π$. Many combinatorial properties can be seen as monotone properties. The problem of finding a minimum subset of $U$ satisfying $Π$ is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to \emph{enumerate} multiple small solutions rather than to \emph{find} a single small solution. To this end, given a weight function $w: U \to \mathbb N$ and an integer $k$, we devise algorithms that \emph{approximately} enumerate all minimal subsets of $U$ with weight at most $k$ satisfying $Π$ for various monotone properties $Π$, where "approximate enumeration" means that algorithms output all minimal subsets satisfying $Π$ whose weight at most $k$ and may output some minimal subsets satisfying $Π$ whose weight exceeds $k$ but is at most $ck$ for some constant $c \ge 1$. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most $k$ with constant approximation factors.