Does Trump have the power to send National Guard troops to the border?

Yea, why wouldn't he?

Yes, President Donald Trump certainly has the authority to call of National Guard forces in the defense of our southern border which is at risk from terrorists - radical Muslim, narco-terrorists, MS-13, etc. CW5 Jack Cardwell US Title 10 and US Title 32 are both Federal statutes applicable to calling up National Guard forces for specified purposes in CONUS.
I am posting a couple of the applicable statutes from 32 US Code
32 U.S. Code § 902 - Homeland defense activities: funds
The Secretary of Defense may provide funds to a Governor to employ National Guard units or members to conduct homeland defense activities that the Secretary,[1] determines to be necessary and appropriate for participation by the National Guard units or members, as the case may be.
32 U.S. Code § 904 - Homeland defense duty
(a)Full-Time National Guard Duty.—
All duty performed under this chapter shall be considered to be full-time National Guard duty under section 502(f) of this title. Members of the National Guard performing full-time National Guard duty in the Active Guard and Reserve Program may support or execute homeland defense activities performed by the National Guard under this chapter.
(b)Duration.—
The period for which a member of the National Guard performs duty under this chapter shall be limited to 180 days. The Governor of the State may, with the concurrence of the Secretary of Defense, extend the period one time for an additional 90 days to meet extraordinary circumstances.
(c)Relationship to Required Training.—
A member of the National Guard performing duty under this chapter shall, in addition to performing such duty, participate in the training required under section 502(a) of this title. The pay, allowances, and other benefits of the member while participating in the training shall be the same as those to which the member is entitled while performing the duty under this chapter. The member is not entitled to additional pay, allowances, or other benefits for participation in training required under section 502(a)(1) of this title.
(d)Readiness.—To ensure that the use of units and personnel of the National Guard of a State for homeland defense activities does not degrade the training and readiness of such units and personnel, the following requirements shall apply in determining the homeland defense activities that units and personnel of the National Guard of a State may perform:
(1)The performance of the activities is not to affect adversely the quality of that training or otherwise interfere with the ability of a member or unit of the National Guard to perform the military functions of the member or unit.
(2)The performance of the activities is not to degrade the military skills of the members of the National Guard performing those activities.
FYI COL Mikel J. Burroughs LTC Stephen C. LTC Orlando Illi LTC (Join to see) LTC Ivan Raiklin, Esq. Maj Bill Smith, Ph.D. Capt Seid Waddell Capt Jeff S. CPT Jack Durish MSG Dan Walther MSgt Robert C Aldi SFC Stephen King MSgt Danny Hope SGT Gregory Lawritson Cpl Craig Marton SP5 Mark Kuzinski SGT (Join to see) Maj Marty Hogan

Please allow me to elaborate. The article is correct but it doesn't really tell the full story. Yes, the PCA does not follow the Guard in a T32 status. And usually, as long as the Federal government is paying the bill governors will send their troops to the border. Hey, who is going to decline free training for your troops? Oh yea- a governor that wants to make a political point at the expense of his Guardsmen. So back to the rest of the story. So what do Guardsmen on the border do? The exact same thing the Boy Scouts of America can do. Watch and report. We all know that intelligence drives operations. Guess what you can't use while your sitting your butt on the border? You can't use federal intelligence assets. All that stuff your commander gets to conduct almost any mission is off limits for two reasons. The first, your intel guys are not allowed to report on anybody who is on US soil. That includes illegal aliens. That is based on Executive Order 12333. Second, all that cool equipment that your MEB has assigned to it to gather all that SIGNT, GEOINT, all other INTs are not allowed to be used on a T32 status. So good luck waisting millions of US treasure for no real tangible benefit. Just saying.