We can't know how much influence any particular batting coach has on his team's hitting success or lack thereof. But we can know that the Mets' hitting as a team during Bill Robinson's six seasons as the team's batting coach was far, far beyond anything that the Mets had seen before or have seen since.
OPS+ is baseball-reference.com's stat that shows OPS performance relative to the league and adjusted for home park effects. A team OPS+ of 100 for a season means the team OPS was exactly league average when you adjust for the ease or difficulty of scoring runs in the team's home park.
In the Mets' 45 season history 1962-2006, the highest OPS+ the franchise has ever achieved is a 117 in 1988, during Robinson's term as hitting coach. Two seasons are tied for the second and third highest OPS+ in franchise history, just a tiny fraction below the top year, at 116. Both those 116 OPS+ seasons were also Bill Robinson years: 1986 and 1987. So three of Bill's 6 years were the three best hitting years in Mets franchise history using OPS+. Did Bill just inherit this talent? Well 1983, the year immediately before Robinson arrived the Mets team OPS+ was a horrible 84. In Bill's very first year the Mets immediately zoomed to 101, above average - this was just the 2nd the Mets as a team had ever been above 100 in OPS+ in what was then the 22 history of the franchise. In his next season, 1985, the Mets went to 105, a new franchise record, then in the next three years 116, 116, 117 and finally in Robinson's last season, down to 109. That "disappointing" drop to 109 still remains better than all but one season in Met history (1999: OPS+ 112) in which the Mets' batting coach was someone other than Bill Robinson. In the six years immediately before Bill became the hitting coach, the Mets' average OPS+ was 92, in the 6 seasons after Robinson left it was 97, but in the six seasons he was the hitting coach the Mets average OPS+ was over 110, closer to 111 -- again, that six year average was better than all but one single season in all the other seasons in Mets franchise history.
In short, from a hitting point of view, Mets franchise history can be divided into two periods -- the Golden Age during which Bill Robinson was the hitting coach, and all the other seasons when he wasn't. How much credit he should get for that I don't know, but to whatever extent the facts speak for themselves they certainly speak loudly.