As was seen with Enigma, if there are enough possible keys, then a coding system will be safe since no-one has the time or computational power to find the keys quick enough. Enigma was only broken by short cuts and clever tricks by the cryptanalysists, who used weaknesses in the implementation of Enigma to solve the coded messages. If mistakes were not made then a huge number of possible solutions could ensure a coding system was unbreakable in practice.
As computers' power increases, a bigger and bigger number of keys are needed to guarantee safety of cryptography. The most popular systems currently used, Triple DES and IDEA, provide this.
The only major problem with Enigma and all the codes that proceeded it were that the operators had to have a book of keys for each day, or each message. Distributing these codes was possible but awkward when it needed to go to all Enigma operators in the war, but trying to distribute a book for people to use when accessing the Internet nowadays is laughable. Without a secure way of passing around keys, cryptography for the general public is not possible.
The problem has been solved by the biggest breakthrough for cryptography, some would argue, since the monoalphabetic cypher: By asymmetric cyphers such as the most popular RSA.