Snippet from https://betterprogramming.pub/upload-files-to-next-js-with-api-routes-839ce9f28430

upload_api_v3.ts

import nextConnect from 'next-connect';
import multer from 'multer';

// Returns a Multer instance that provides several methods for generating 
// middleware that process files uploaded in multipart/form-data format.
const upload = multer({
  storage: multer.diskStorage({
    destination: './public/uploads',
    filename: (req, file, cb) => cb(null, file.originalname),
  }),
});

const apiRoute = nextConnect({/* ... */});

// Returns middleware that processes multiple files sharing the same field name.
const uploadMiddleware = upload.array('theFiles');

// Adds the middleware to Next-Connect
apiRoute.use(uploadMiddleware);

apiRoute.post((req, res) => {
...