Could Congress have used budget reconciliation to fund a border wall and avoid the current shutdown?
Reconciliation is only useful for reducing the vote threshold of a bill to a bare majority. Congress is currently split so reconciliation is unusable, there is no way for either party to assemble a simple majority in both chambers of congress.
The shut down started during the last congress (while Republicans controlled both chambers). The (Republican controlled) house rejected the continuing resolution after it passed the Senate because of Trump demanding the wall funding. In the new congress the (now Democratic) house has passed a continuing resolution, but Mitch McConnell has refused to put it up for a vote in the Senate because he wants to avoid the optics of forcing his members to this vote. There are currently enough votes in the Senate (given the public statements of some Republican Senators) to pass the continuing resolution, but there probably aren't enough votes for the veto override that will be needed after (assuming Trumps vetoes anything without his wall funding).