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President Trump recently tweeted that he may close the country’s southern border if Congress does not appropriate the funds to build a border wall.

... We build a Wall or close the Southern Border ...

^{– @realDonaldTrump (Tweet 1 and Tweet 2)}

So, does the executive branch have the legal authority to either partially or fully close the country’s borders?

I would say YES.

The president has the clear legal authority to "suspend the entry of all aliens" if he/she finds the entry of aliens "detrimental to the interests of the United States".

United States Code > Title 8 > Chapter 12 > Section 1182(f) "Inadmissable Aliens" reads:

Suspension of entry or imposition of restrictions by President

Whenever the President finds that the entry of any aliens or of any
class of aliens into the United States would be detrimental to the
interests of the United States, he may by proclamation, and for such
period as he shall deem necessary, suspend the entry of all aliens or
any class of aliens as immigrants or nonimmigrants, or impose on the
entry of aliens any restrictions he may deem to be appropriate.

In 2017, the U.S. Supreme Court affirmed this authority in Trump v Hawaii.

Here's an excerpt from Chief Justice John Roberts' majority opinion:

By its terms, §1182(f) exudes deference to the President in every
clause. It entrusts to the President the decisions whether and when to
suspend entry (“[w]henever [he] finds that the entry” of aliens “would
be detrimental” to the national interest); whose entry to suspend
(“all aliens or any class of aliens”); for how long (“for such period
as he shall deem necessary”); and on what conditions (“any
restrictions he may deem to be appropriate”).

It is therefore unsurprising that we have previously observed that
§1182(f) vests the President with “ample power” to impose entry
restrictions in addition to those elsewhere enumerated in the INA
(finding it “perfectly clear” that the President could “establish a
naval blockade” to prevent illegal migrants from entering the United
States).

The sole prerequisite set forth in §1182(f) is that the President
“find” that the entry of the covered aliens “would be detrimental to
the interests of the United States.”