Individual trapped atomic qubits represent one of the most promising technologies to scale quantum computers, owing to their low idle errors and the ability to implement a full set of reconfigurable gate operations via focused optical fields. However, the fidelity of quantum gate operations can be limited by weak confinement of the atoms transverse to the laser. We present measurements of this effect by performing individually addressed entangling gates in chains of up to 25 trapped atomic ions. We present a semiclassical model that accurately describes the observed decoherence from the residual heating of the ions caused by noisy electric fields. We suppress these effects by cotrapping ancilla ions for sympathetically cooling the qubit ions throughout a quantum circuit.