WooCommerce - Use "Starting at" prefix for variable price range

functions.php

<?php
/**
 * Custom variable price HTML.
 * Shows "Starting at" prefix with when min price is different from max price.
 *
 * @param stirng $price Product price HTML
 * @param WC_Product_Variable $product Product data.
 * @return string
 */
function cs_my_wc_custom_variable_price_html( $price, $product ) {
	$prices = $product->get_variation_prices( true );

	$min_price = current( $prices['price'] );
	$max_price = end( $prices['price'] );

	// Return price if min is equal to max.
	if ( $min_price === $max_price || $product->is_on_sale() ) {
		return $price;
	}

	return sprintf( __( 'Starting at %s', 'your-text-domain' ), wc_price( $min_price ) . $product->get_price_suffix() );
}

add_filter( 'woocommerce_variable_price_html', 'cs_my_wc_custom_variable_price_html', 10, 2 );

/**
 * Custom variable sale price HTML.
 * Shows "Starting at" prefix with when min price is different from max price.
 *
 * @param stirng $price Product price HTML
 * @param WC_Product_Variable $product Product data.
 * @return string
 */
function cs_my_wc_custom_variable_sale_price_html( $price, $product ) {
	$prices = $product->get_variation_prices( true );

	$min_price = current( $prices['price'] );
	$min_regular_price = current( $prices['regular_price'] );
	$max_regular_price = end( $prices['regular_price'] );

	// Return price if min is equal to max.
	if ( $min_regular_price === $max_regular_price ) {
		return $price;
	}

	$price = $product->get_price_html_from_to( $min_regular_price, $min_price );

	return sprintf( __( 'Starting at %s', 'your-text-domain' ), $price . $product->get_price_suffix() );
}

add_filter( 'woocommerce_variable_sale_price_html', 'cs_my_wc_custom_variable_sale_price_html', 10, 2 );

woocommerce_variable_sale_price_html hook is deprecated and not used anymore, if you wanna display a prefix and also display previous price when it's on sale then use the following:

add_filter( 'woocommerce_variable_price_html', 'custom_min_max_variable_price_html', 10, 2 );
function custom_min_max_variable_price_html( $price, $product ) {
    $prices = $product->get_variation_prices( true );
    $min_price = current( $prices['price'] );

    $min_keys = current(array_keys( $prices['price'] ));
    $min_price_regular = $prices['regular_price'][$min_keys];
    $min_price_html = wc_price( $min_price ) . $product->get_price_suffix();

    if( $min_price_regular != $min_price ){ // When min price is on sale (Can be removed)
        $min_price_regular_html = '<del>' . wc_price( $min_price_regular ) . $product->get_price_suffix() . '</del>';
        $min_price_html = $min_price_regular_html .'<ins>' . $min_price_html . '</ins>';
    }
    $price = sprintf( __( 'From %1$s', 'woocommerce' ), $min_price_html );

    return $price;
}