The utilization of synthetic data for fingerprint recognition has garnered increased attention due to its potential to alleviate privacy concerns surrounding sensitive biometric data. However, current methods for generating fingerprints have limitations in creating impressions of the same finger with useful intra-class variations. To tackle this challenge, we present GenPrint, a framework to produce fingerprint images of various types while maintaining identity and offering humanly understandable control over different appearance factors such as fingerprint class, acquisition type, sensor device, and quality level. Unlike previous fingerprint generation approaches, GenPrint is not confined to replicating style characteristics from the training dataset alone: it enables the generation of novel styles from unseen devices without requiring additional fine-tuning. To accomplish these objectives, we developed GenPrint using latent diffusion models with multimodal conditions (text and image) for consistent generation of style and identity. Our experiments leverage a variety of publicly available datasets for training and evaluation. Results demonstrate the benefits of GenPrint in terms of identity preservation, explainable control, and universality of generated images. Importantly, the GenPrint-generated images yield comparable or even superior accuracy to models trained solely on real data and further enhances performance when augmenting the diversity of existing real fingerprint datasets.