Is Trade Skepticism on the Rise?

Perhaps, but it has a long history, as shown by this AEI paper that looks at public responses to trade-related polling over time.  Here's one illustration:

Do you think it should be the policy of the United States to restrict foreign imports into this country in order to protect American industry and American jobs, or do you think there should be no restrictions on the sale of foreign products in the United States in order to permit the widest choice and the lowest prices for the American consumer?

Restrict

No

Imports

Restrictions

To protect

On sale

American Industry

Of foreign

Not Sure

And Jobs

Products

(Vol.)/Refused

Jul. 4-8, 1982

LAT

72

22

6

May 8-12, 1983

LAT

68

26

6

Feb. 4-9, 1984

LAT

67

26

7

Apr. 28-May 3, 1984*

LAT

65

25

10

Sep. 21-26, 1985

LAT

65

27

8

Oct. 28-Nov. 2, 1987

LAT

69

21

10

Jul. 5-10, 1988

LAT

72

21

7

Sep. 23-25, 1988+

LAT

71

19

10

Sep. 25-26, 1988^

LAT

66

26

8

Sep. 29, 1988^

LAT

68

21

11

Oct. 11-13, 1988#

LAT

71

21

8

Oct. 13-15, 1988%

LAT

73

18

9

Jan. 8-12, 1991

LAT

67

25

8

Jan. 31-Feb. 3, 1992

LAT

70

23

7

Jun. 12-14, 1993

LAT

68

24

8

Aug. 3-6, 1996

LAT

63

28

9

Sep. 6-9, 1997

LAT

67

24

9

More historical data, with a number of variations of the question, in the paper.