It seems to me that the best way to approach this question is to look at what the Christians at the time of the apostles did. If they worshipped Jesus then we should probably do so too. If they did not worship Jesus then we should not.
In recent book entitled Did the first Christians worship Jesus? (SPCK, 2010), Professor James Dunn has explored the NT evidence for the practice of early Christians. This includes a detailed discussion of the worship-words used in the NT. It is worth summarising his conclusions.
Worship-words applied only to God
latreuein (lit. “to serve”) – denotes cultic worship
sebein (lit. “to worship”) – denotes worship of a god, perhaps including attachment
ainein (lit. “to praise”) – denotes praise of God (as opposed to epainein which is used of praise of other individuals)
eucharistein (lit. “to give thanks”) – in the context of prayer, denotes giving thanks (to God) (Note: in other contexts eucharistein is used of humans; cf. Luke 17:16, Rom 16:4)
eulogetos (lit. “blessed”) – used in benediction of God
proseuchesthai (lit. “to pray”) – prayer to a god
Worship-words applied to Jesus
proskynein (lit. “to bow”) – denotes homage before a monarch or a superior
epikaleisthai (lit. “to call upon”) – denotes appeal or innovocation (e.g. prayer)
phobein (lit. “to fear”) – in a religious context denotes reverence (Eph 5:21)
charis (lit. “grace”) – can be used to express gratitude, sometimes in the form of prayer (1 Tim 1:12)
doxazein (lit. “to glory”) – used in variety of contexts of honouring someone
deesthai (lit. “to ask”) – used in variety of contexts of making requests of someone
parakalein (lit. “to urge”) – as well as everyday sense of urging, can be used of requests made in prayer (2 Cor 12:8-9)
From this survey of worship-words we can detect a distinct pattern. Some words are reserved for God alone. These words are usually those with strong religious connotations. Other words are used of Jesus. These are usually those which can be used in a variety of contexts, though some have a religious aspects and are used as such of Jesus. This indicates that the early Christians worshipped God (as God) and honoured Jesus (as Lord).
So there is a sense in which we should reverence Jesus. We should offer him honour and praise, and can approach him in prayer to express our gratitude and even to appeal to him for help. However, like the early Christians, we should reserve a special worship for God alone, to whom we offer praise, blessing and our service, and to whom we make our prayers.
The Greek Old Testament naturally forbids the Jews to proskynein (lit. “to bow”) to idols or “other gods” (Exodus 20:5 οὐ προσκυνήσεις αὐτοῖς ; Exodus 34:14 οὐ γὰρ μὴ προσκυνήσητε θεῷ ἑτέρῳ ). But does not forbid bowing down to monarchs.