All of the ways characters went from one universe to another throughout the series.
Olivia's Natural Ability
Olivia is naturally able to hop between universes. Cortexiphan, developed by Walter Bishop and William Bell, prevents Olivia's brain from undergoing a natural limitation process that would prevent her from using this ability. Use of her ability does not seem to negatively effect either universe.
William Bell pulled Olivia into the Redverse before she knew how to use her own powers. The mechanism by which he did this is unknown. Olivia was driving her car when the switch happened. It is unknown if this method of transfer adversely affects either universe, but there is no good reason to think it doesn't.
Walter made a device in 1985 to cross into the Redverse to deliver a cure to Peter. Using this device to rip a hole between universes was the first crack in a series of cracks that led to the deterioration of both universes, particularly the Redverse.
Due to each universe vibrating at its own frequency, harmonic rods could be used to tune a subject to its target universe and initiate a transfer. Three rods form a triangle, with the subject to be moved placed in the center. This method has been used on both people and objects. Something that crosses from universe A to universe B by this method requires something of equal mass to be transferred from universe B to universe A to balance it. If this is not done manually, the universes will naturally pull matter over to balance themselves.
Peter bridged the Blueverse and the Redverse so that anyone (with clearance) could just walk through a couple doorways from one to the other. Not only were there no negative consequences from people crossing over this way, but having the universes connected via this bridge room actually allowed them to start healing.
Observers have the ability to hop from one universe to another. It is unknown if this method of transfer adversely affects either universe, but there is no good reason to think it doesn't. See Observer Abilities for more information.