We should have a system in which workers' rights to organize are fully protected. They should be free to seek unionization -- the most important point being effective protection against retaliatory discharge by anti-union management, although there are other necessary safeguards as well.
If we had that system, then I would be OK with doing without card check. (Card check is a legislative proposal addressing the situation in which management is presented with representation cards signed by a majority of the employees. Under current law, management can demand a secret-ballot election, to ensure that workers' right not to unionize was adequately protected. Card check means that cards signed by the majority would suffice to establish the union as the employees' bargaining representative, without a secret-ballot election.)
We no longer have even an approximation of a fair system, however. The protections of the National Labor Relations Act have become a joke. Companies that fire union organizers can expect no serious consequences, merely a small payment imposed years later. Even that feeble level of protection has been undercut by Republican obstructionism of appointments to the National Labor Relations Board, depriving it of a quorum to do anything at all for long periods.
If Republicans are serious about protecting workers' democratic rights, let them show the same zeal in protecting the right to unionize that they do in protecting the right to reject unionization. Until then, I'm a reluctant supporter of card check. It's not a good solution, but enacting card check in the context of our current dysfunctional system would make that system somewhat more fair, overall.
As an aside, I favor the secret ballot in elections for public office, but it's not a democratic cornerstone in the sense of having been understood as part of democracy from the beginning. At the time of the adoption of the Constitution, voters (white male property owners) would vote by orally announcing their choice in a public forum. The first use of the secret ballot in the United States was in Massachusetts in 1888.