Inter Milan midfielder Eriksen collapsed on the pitch during Denmark's tournament opener against Finland in Copenhagen on June 12 and is currently recovering at home.
Denmark qualified for the knockout stage on a wave of emotion and have become a lot of people's 'second team' after what happened to Eriksen.
But skipper Bale, speaking ahead of the last-16 clash in Amsterdam, said Wales were familiar with being underdogs.
"That's normal anyway. We're always the underdogs and used to that tag. It doesn't make a difference to us," said the Real Madrid forward.
"Obviously we understand the situation that happened with them and we've sent all our best wishes to Christian but it won't have any bearing on the game.
"The players have been in situations before where teams are favourites and supported more.
"Against Turkey they had 18,000 fans in the stadium (when Wales won their group game in Baku 2-0), so it makes no difference."
UEFA's referees chief Roberto Rosetti has praised English official Anthony Taylor for his "perfect" handling of Eriksen's cardiac arrest during the Denmark v Finland match.
Taylor immediately recognised the seriousness of the situation when the Dane collapsed just before half-time in the match on June 12 and signalled for medics to come on within seconds.
"Everyone recognised that Anthony was perfect. He managed this difficult moment in a great way," said Rosetti.
"We are proud about his behaviour, we are proud about his cold blood.
"We recommended to the referees that safety is first, it is the most important target for us.
"The referees must, and they are ready to, stop the match immediately in these situations. Anthony was amazing."
Eriksen has since been discharged from hospital in Copenhagen after being fitted with an implantable cardioverter defibrillator (ICD).
Post of the day
Big decision for Mancini
Italy coach Roberto Mancini finds himself with selection problems of the best kind ahead of their last-16 tie against Austria.
"Verratti or Locatelli? Someone good will not play," was the Italian coach's response when asked about his dilemma.
Matthaus not happy with Low
Matthaus, who featured in five World Cups and four European Championships and won one of each, said Low had created insecurity in the squad with his changes of tactics and personnel.
"I don't have the feeling I had seven years ago when he won the World Cup," he said.
"I was missing the connection between himself on the bench and the players on the field.
"He changed so many things. He changed one player for another and also the system.
"Joshua Kimmich played four positions (against Hungary). How can each player feel confident generally when we are changing so many things during a game? There's no security for the team."
Quote of the day
Wales goalkeeper Danny Ward is well prepared for anything Denmark have to throw at him as he has seen plenty of action already. The Leicester number two, who comes up against the man he plays deputy to in Kasper Schmeichel, is joint-second on the list of 'most saves'.
Wales v Denmark (Round of 16, June 26, Amsterdam, 1700 BST)
Italy v Austria (Round of 16, June 26, London, 2000)