脑袋疼，折腾了个把小时，终于折腾清楚了，如下例所示，selectedIds 是一个字符串数组，axios 传参的时候不需要任何的包装，花括号什么的都不需要

      var that = this
      var selectedIds = this.selectedRows.map((o) => o.id)
      if (selectedIds.length === 0) {
        this.$message.error('请选择需要删除的行！')
        return
      }
      this.$confirm('是否确认删除所选择的行?', '删除', {
        confirmButtonText: '确定', cancelButtonText: '取消', type: 'warning'
      }).then(() => {
        this.axios.post('/Role/DeleteRoles', selectedIds
        ).then(function (response) {
          if (response.data.Success) {
            var d = response.data
            if (d.Success) {
              that.getTableData()
              that.getTreeData()
            }
          }
        })
      }).catch(() => {
        this.$message({
          type: 'info',
          message: '已取消'
        })
      })

Web Api 这边的接收

        [HttpPost]
        public IHttpActionResult DeleteRoles(string[] ids)
        {
            try
            {
                bool result = _services.DeleteRoles(ids);
                if (result)
                    return Json(new { Success = true });
                return Json(new ResponseObj<RoleEntity>
                {
                    Success = false,
                    Msg = "操作失败!",
                    ReplyTime = DateTime.Now
                });
            }
            catch (Exception e)
            {
                return Json(new ResponseObj<RoleVModel>
                {
                    Success = false,
                    Msg = e.Message,
                    ReplyTime = DateTime.Now
                });
            }
        }

没什么特别的直接用一个 string[] 接收就好了