We tested a bunch of FTP clients for iOS devices and found these to be the best. Most of them cost between $1 to $5 but we found the small price to be worth it. We do not endorse the use of any of them, we just tested them all with a starter cloud hosting plan from Thexyz and they worked well.

FTP Client Pro

Seems actively developed with full upload and download via email client and device on iPad and iPhone. View text files (doc, txt, rtf, pdf, pps, xls, etc). Edit simple text files (txt, css, php, xml, html, csv, htm, etc)

FTP Client Pro - LessIsMore Development

Easy FTP Pro

Easy FTP has a nice text editor which editing documents with the SFTP client for iPhone/iPod Touch.

Easy FTP Pro - Juan Ramon Rivero

iFiles

A nice app that works really well with the iOS, all the features you would expect are also available.

iFiles - Imagam.com

FTP Sprite+

FTP Sprite can turn your iPhone, ipad, ipod into ftp client, download files from ftp server and upload files into ftp server.

FTP Sprite+ - ujweng

Phone Drive

Optimized for iOS 7 : Fully redesigned to embrace the fresh new look and feel.

Phone Drive - Eightythree Technology

FTP Browser

FTP Browser is currently the only iPhone, iPod, or iPad ftp web browser available.

FTP Browser - Brock Haymond

iTransfer

Upload and Download files to your device via FTP/FTPS/SFT.

iTransfer - File Upload / File Download Tool - ComcSoft