There certainly was concern that Nelson Rockefeller's wealth would compromise him in certain ways. Republicans tended to not trust his more left-wing views, while Democrats were wary of the fact that his wealth and connections presented a conflict of interest waiting to happen. It was brought up it different ways during his hearings. Courtesy of the Ford Library, a series of correspondence in regards to Nelson's appointment and hearings:

https://www.fordlibrarymuseum.gov/library/document/0204/1511932.pdf

On page 27, as part of an AP report:

"HARRIS POLL NEW YORK CAP> -- POLLSTER LOUIS HARRIS SAID TODAY THAT IN TWO MONTHS THE PUBLIC HAS SWITCHED FROM 55-31 APPROVAL TO 43-39 DISAPPROVAL OF FORMER NEW YORK GOV~ NELSON ROCKEFELLER'S APPOINTMENT AS VICE PRESIDENT.

HARRIS SAID DISAPPROVAL OF $2 IN MILLION GIFTS AND LOANS ROCKEFELLER MADE TO APPOINTEES, AND FINANCING OF A BOOK UNCOMPLIMENTARY TO HIS 1970 GUBERNATORIAL OPPONENT, ARTHUR GOLDBERG, HAVE BEEN THE MAIN FACTORS COSTING ROCKEFELLER PUBLIC SUPPORT.

IN THE LATEST SURVEY OF 1,525 PERSONS ON THE SUBJECT, CONDUCTED ~OV. 1 THROUGH 5, A 43-39 PER CENT PLURALITY DISAPPROVED OF ROCKEFELLER'S DESIGNATION BY PRESIDENT FORD, WITH 18 PER CENT LISTED AS NOT SURE.

IN A POLL JUST AFTER THE APPOINTMENT IN EARLY SEPTEMBER, THOSE SURVEYED VOICED APPROVAL BY 55 TO 31 PER CENT, WITH 14 PER CENT UNSURE. A LARGE MAJORITY, 71 TO 9 PER CENT, EXPRESSED THE BELIEF THAT THE ROCKEFELLER FAMILY HAD ''BEEN GENEROUS IN GIVING THEIR MONEY TO MANY GOOD CAUSES,'' BUT BY 47 TO 34 PER CENT THEY FELT THE FAMILY'S VAST WEALTH WOULD CREATE A CONFLICT OF INTEREST IF HE WERE CONFIRMED AS VICE PRESIDENT."

Bold is emphasis mine, but I included the whole passage for context. It was also revealed that Rockefeller had used his money to give political gifts, most famously to Henry Kissinger, and to support a book attacking Arthur J. Goldberg, an opponent to Nelson in the 1970 Gubernatorial election. Robert Byrd specifically questioned any sort of conflict of interest in the hearings:

“One cannot explode a myth that is not a myth,” stated Byrd, as the second day of hearings opened Sept. 24. “It's not a myth….The name of Rockefeller is known to every person in America and associated with it is the meaning of tremendous wealth….To say that it's a myth does not dissipate the fact.

"I wonder if you yourself may be conscious of the power that the two [great political power and extensive economic influence] when combined really add up to….I wonder if you can separate the interest of big business and the national interest when they diverge," speculated Byrd.

After considerable debate between the two concerning the existence of the Rockefellers' economic influence, Rockefeller finally responded, “yes sir—no problem!” to Byrd's central question."

https://library.cqpress.com/cqalmanac/document.php?id=cqal74-1223238#926

So to answer your question, it was an issue to both officials and the general public. I couldn't tell you how frequently it was talked about or how much influence it had over political proceedings, but it was an issue.