Racing 92's Arundell, 21, signed a two-year contract extension with the Parisians after rejecting a move to Bath that would also have included one of the Rugby Football Union's 25 hybrid contracts.
It means Arundell, who scored five tries in the Rugby World Cup match against Chile in September, cannot play for England under existing regulations.
With three English Premiership clubs -- Wasps, Worcester and London Irish -- all going out of business last season, some observers argue the rule is no longer fit for purpose.
"How will we promote our competition as being at a very high level if you're wide open to the best players playing outside the country?" Baxter said.
"That will never help promote the Premiership and without promoting the Premiership I don't think you'll ever get a successful England side.
"The best way to keep young players in this country is by letting them know that staying in this country gives them the best opportunity to play international rugby."
New Zealand is the only other country to have a ban on fielding overseas players for the national team.
Exeter host Munster in the European Champions Cup on Sunday, with the English side looking to follow-up their opening round win over Toulon -- one of seven victories in eight games for Premiership sides in the competition.
Those successes came against a backdrop of concern regarding Premiership clubs' ability to compete in Europe when operating with a reduced salary cap, while the likes of Arundell and fellow England internationals Jack Willis, David Ribbans and Joe Marchant have all opted to join French sides.
Former Exeter players such as winger Jack Nowell, fly-half Joe Simmonds and No 8 Sam Simmonds left the Chiefs for French clubs this season.
"Because of the negativity surrounding the Premiership lately, it would be very easy to take the Premiership clubs lightly. Maybe that's what happened," said Baxter.
"We should be talking positively about the Premiership and the results at the weekend bear that out.
"English teams will be competitive, our challenge going forward is how we maintain that."
