Using static routes come with a couple of disadvantages in this scenario:
First thing to consider, they are not aware of upstream failure thus the router may continue to send packets to its peer when in fact the peer is no longer there. Admittedly this can be mitigated somewhat by using a connected route which work well in situations where the line status changes. But consider that that not all WAN transmission medium failure results in a change in interface status (frame relay is a classic example). In addition, you may have a more remote failure rather than between your router and the one it peers with. For example, an ISP's main peering router may go down and stop participating in BGP. Thus any of its customer peering routers will become aware of this failure as they will also lose these routes. But in a static routing environment, your router will continue to send traffic to the ISP neighbor even tho this ISP router has lost all its BGP routes from its upstream peer and may not have an alternative route - in a properly-designed ISP network however this should never be the case.
All that said, Cisco are continually improving on the traditional ideas. Now they allow you to "monitor" a remote node and to use a static route to that node if it available. If that node disappears, the router can be made to fallback to a different static route. Cisco call it "reliable static routing" and it uses ICMP echo's to determine the status of a remote node. Albeit this is not infallible as I once had a situation where an ISP introduced a firewall between my router and the node I was polling. This firewall didn't allow ICMP pings which made things interesting for awhile!
Second thing to consider, static routing may introduce a significant administrative overhead regarding having to update your routing whenever the network is changed, expanded etc.
